ring (algebra)
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, rings are
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s that generalize
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
: multiplication need not be
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
s need not exist. In other words, a ''ring'' is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
equipped with two
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
s satisfying properties analogous to those of
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. Ring elements may be numbers such as
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, but they may also be non-numerical objects such as
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s,
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
, functions, and
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. Formally, a ''ring'' is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
, is distributive over the addition operation, and has a multiplicative
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior.
Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, the theory of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s, is a major branch of
ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
. Its development has been greatly influenced by problems and ideas of
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. The simplest commutative rings are those that admit division by non-zero elements; such rings are called
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
. Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
of an
affine algebraic variety Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
, and the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
of a number field. Examples of noncommutative rings include the ring of real
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
with ,
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
s in
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
,
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of ...
s in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, rings of differential operators, and
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
s in
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by
Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
,
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, Fraenkel, and Noether. Rings were first formalized as a generalization of
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s that occur in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and of
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
s and rings of invariants that occur in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
. They later proved useful in other branches of mathematics such as
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.


Definition

A ring is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''R'' equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms # ''R'' is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
under addition, meaning that: #* (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, + is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
). #* ''a'' + ''b'' = ''b'' + ''a'' for all ''a'', ''b'' in ''R''   (that is, + is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
). #* There is an element 0 in ''R'' such that ''a'' + 0 = ''a'' for all ''a'' in ''R''   (that is, 0 is the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from element ...
). #* For each ''a'' in ''R'' there exists −''a'' in ''R'' such that ''a'' + (−''a'') = 0   (that is, −''a'' is the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
of ''a''). # ''R'' is a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
under multiplication, meaning that: #* (''a'' ⋅ ''b'') ⋅ ''c'' = ''a'' ⋅ (''b'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, ⋅ is associative). #* There is an element 1 in ''R'' such that and for all ''a'' in ''R''   (that is, 1 is the
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
). # Multiplication is distributive with respect to addition, meaning that: #* ''a'' ⋅ (''b'' + ''c'') = (''a'' ⋅ ''b'') + (''a'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R''   (left distributivity). #* (''b'' + ''c'') ⋅ ''a'' = (''b'' ⋅ ''a'') + (''c'' ⋅ ''a'') for all ''a'', ''b'', ''c'' in ''R''   (right distributivity).


Notes on the definition

In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a rng (IPA: ). For example, the set of
even integer In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ ...
s with the usual + and ⋅ is a rng, but not a ring. As explained in ' below, many authors apply the term "ring" without requiring a multiplicative identity. The multiplication symbol ⋅ is usually omitted; for example, ''xy'' means . Although ring addition is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
, ring multiplication is not required to be commutative: ''ab'' need not necessarily equal ''ba''. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called ''
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s''. Books on commutative algebra or algebraic geometry often adopt the convention that ''ring'' means ''commutative ring'', to simplify terminology. In a ring, multiplicative inverses are not required to exist. A non
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
commutative ring in which every nonzero element has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
is called a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: .) Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative. For these authors, every
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
is a "ring".


Illustration

The most familiar example of a ring is the set of all integers \mathbf, consisting of the
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s : ... , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.


Some properties

Some basic properties of a ring follow immediately from the axioms: * The additive identity is unique. * The additive inverse of each element is unique. * The multiplicative identity is unique. * For any element ''x'' in a ring ''R'', one has (zero is an
absorbing element In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element i ...
with respect to multiplication) and . * If in a ring ''R'' (or more generally, 0 is a unit element), then ''R'' has only one element, and is called the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for a ...
. * If a ring ''R'' contains the zero ring as a subring, then ''R'' itself is the zero ring. * The
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
holds for any ''x'' and ''y'' satisfying .


Example: Integers modulo 4

Equip the set \mathbf/4\mathbf = \left\ with the following operations: * The sum \overline + \overline in Z/4Z is the remainder when the integer is divided by 4 (as is always smaller than 8, this remainder is either or ). For example, \overline + \overline = \overline and \overline + \overline = \overline. * The product \overline \cdot \overline in Z/4Z is the remainder when the integer ''xy'' is divided by 4. For example, \overline \cdot \overline = \overline and \overline \cdot \overline = \overline. Then Z/4Z is a ring: each axiom follows from the corresponding axiom for Z. If ''x'' is an integer, the remainder of ''x'' when divided by 4 may be considered as an element of Z/4Z, and this element is often denoted by or \overline, which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any \overline in Z/4Z is \overline. For example, -\overline = \overline = \overline.


Example: 2-by-2 matrices

The set of 2-by-2
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
with entries in a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
is :\operatorname_2(F) = \left\. With the operations of matrix addition and
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
, \operatorname_2(F) satisfies the above ring axioms. The element \left( \begin 1 & 0 \\ 0 & 1 \end\right) is the multiplicative identity of the ring. If A = \left( \begin 0 & 1 \\ 1 & 0 \end \right) and B = \left( \begin 0 & 1 \\ 0 & 0 \end \right), then AB = \left( \begin 0 & 0 \\ 0 & 1 \end \right) while BA = \left( \begin 1 & 0 \\ 0 & 0 \end \right); this example shows that the ring is noncommutative. More generally, for any ring , commutative or not, and any nonnegative integer , the square matrices of dimension with entries in form a ring: see
Matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
.


History


Dedekind

The study of rings originated from the theory of
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
s and the theory of
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s. In 1871,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.


Hilbert

The term "Zahlring" (number ring) was coined by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if then , , , , , and so on; in general, ''a''''n'' is going to be an integral linear combination of 1, ''a'', and ''a''2.


Fraenkel and Noether

The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every
non-zero-divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
to have a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
. In 1921,
Emmy Noether Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper ''Idealtheorie in Ringbereichen''.


Multiplicative identity and the term "ring"

Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not. Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2006 that use the term without the requirement for a 1. Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that the natural notion for rings is the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
rather than the direct sum. He further argues that rings without a multiplicative identity are not totally associative (the product of any finite sequence of ring elements, including the empty sequence, is well-defined, independent of the order of operations) and writes "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1". Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention: :* to include a requirement a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit", or "ring with 1". :* to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.


Basic examples


Commutative rings

* The prototypical example is the ring of integers with the two operations of addition and multiplication. * The rational, real and complex numbers are commutative rings of a type called
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
. * A unital associative
algebra over a commutative ring In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
is itself a ring as well as an -module. Some examples: ** The algebra of polynomials with coefficients in . ** The algebra of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
with coefficients in . ** The set of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
real-valued functions defined on the real line forms a commutative -algebra. The operations are
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
addition and multiplication of functions. ** Let be a set, and let be a ring. Then the set of all functions from to forms a ring, which is commutative if is commutative. The ring of continuous functions in the previous example is a subring of this ring if is the real line and . * The ring of
quadratic integers In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and (usual) integers. When algebra ...
, the integral closure of \mathbf in a quadratic extension of \mathbf. It is a subring of the ring of all
algebraic integers In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
. * The ring of
profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \math ...
s \widehat, the (infinite) product of the rings of ''p''-adic integers \mathbf_p over all prime numbers ''p''. * The Hecke ring, the ring generated by Hecke operators. * If is a set, then the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of becomes a ring if we define addition to be the
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
of sets and multiplication to be intersection. This is an example of a
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
.


Noncommutative rings

* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n''
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For , this matrix ring is isomorphic to ''R'' itself. For (and ''R'' not the zero ring), this matrix ring is noncommutative. * If ''G'' is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, then the
endomorphisms In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a grou ...
of ''G'' form a ring, the
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
End(''G'') of ''G''. The operations in this ring are addition and composition of endomorphisms. More generally, if ''V'' is a
left module In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
over a ring ''R'', then the set of all ''R''-linear maps forms a ring, also called the endomorphism ring and denoted by End''R''(''V''). *The
endomorphism ring of an elliptic curve In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero. * If ''G'' is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
and ''R'' is a ring, the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
of ''G'' over ''R'' is a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
over ''R'' having ''G'' as basis. Multiplication is defined by the rules that the elements of ''G'' commute with the elements of ''R'' and multiply together as they do in the group ''G''. * The
ring of differential operators In mathematics, a differential operator is an Operator (mathematics), operator defined as a function of the derivative, differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation ...
(depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s are noncommutative.


Non-rings

* The set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s . with the usual operations is not a ring, since is not even a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
(the elements are not all
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers . The natural numbers (including 0) form an algebraic structure known as a
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
(which has all of the axioms of a ring excluding that of an additive inverse). * Let ''R'' be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
: (f * g)(x) = \int_^\infty f(y)g(x - y) \, dy. Then ''R'' is a rng, but not a ring: the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
has the property of a multiplicative identity, but it is not a function and hence is not an element of ''R''.


Basic concepts


Products and powers

For each nonnegative integer , given a sequence (a_1,\ldots,a_n) of elements of , one can define the product \textstyle P_n = \prod_^n a_i recursively: let and let for . As a special case, one can define nonnegative integer powers of an element of a ring: and for . Then for all .


Elements in a ring

A left
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. A right zero divisor is defined similarly. A
nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
is an element a such that a^n = 0 for some n > 0. One example of a nilpotent element is a
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear tr ...
. A nilpotent element in a nonzero ring is necessarily a zero divisor. An
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
e is an element such that e^2 = e. One example of an idempotent element is a
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
in linear algebra. A
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
is an element a having a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
; in this case the inverse is unique, and is denoted by a^. The set of units of a ring is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under ring multiplication; this group is denoted by R^\times or R^* or U(R). For example, if ''R'' is the ring of all square matrices of size ''n'' over a field, then R^\times consists of the set of all invertible matrices of size ''n'', and is called the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
.


Subring

A subset ''S'' of ''R'' is called a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
if any one of the following equivalent conditions holds: * the addition and multiplication of ''R''
restrict In the C programming language, restrict is a keyword, introduced by the C99 standard, that can be used in pointer declarations. By adding this type qualifier, a programmer hints to the compiler that for the lifetime of the pointer, no other p ...
to give operations ''S'' × ''S'' → ''S'' making ''S'' a ring with the same multiplicative identity as ''R''. * 1 ∈ ''S''; and for all ''x'', ''y'' in ''S'', the elements ''xy'', ''x'' + ''y'', and −''x'' are in ''S''. * ''S'' can be equipped with operations making it a ring such that the inclusion map ''S'' → ''R'' is a ring homomorphism. For example, the ring Z of integers is a subring of the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of real numbers and also a subring of the ring of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s Z 'X''(in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2Z does not contain the identity element 1 and thus does not qualify as a subring of Z; one could call 2Z a subrng, however. An intersection of subrings is a subring. Given a subset ''E'' of ''R'', the smallest subring of ''R'' containing ''E'' is the intersection of all subrings of ''R'' containing ''E'', and it is called ''the subring generated by E''. For a ring ''R'', the smallest subring of ''R'' is called the ''characteristic subring'' of ''R''. It can be generated through addition of copies of 1 and −1. It is possible that n\cdot 1=1+1+\ldots+1 (''n'' times) can be zero. If ''n'' is the smallest positive integer such that this occurs, then ''n'' is called the '' characteristic'' of ''R''. In some rings, n\cdot 1 is never zero for any positive integer ''n'', and those rings are said to have ''characteristic zero''. Given a ring ''R'', let \operatorname(R) denote the set of all elements ''x'' in ''R'' such that ''x'' commutes with every element in ''R'': xy = yx for any ''y'' in ''R''. Then \operatorname(R) is a subring of ''R'', called the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of ''R''. More generally, given a subset ''X'' of ''R'', let ''S'' be the set of all elements in ''R'' that commute with every element in ''X''. Then ''S'' is a subring of ''R'', called the
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
(or commutant) of ''X''. The center is the centralizer of the entire ring ''R''. Elements or subsets of the center are said to be ''central'' in ''R''; they (each individually) generate a subring of the center.


Ideal

Let ''R'' be a ring. A left ideal of ''R'' is a nonempty subset ''I'' of ''R'' such that for any ''x'', ''y'' in ''I'' and ''r'' in ''R'', the elements x+y and rx are in ''I''. If R I denotes the ''R''-span of ''I'', that is, the set of finite sums :r_1 x_1 + \cdots + r_n x_n \quad \textrm\;\textrm\; r_i \in R \; \textrm \; x_i \in I, then ''I'' is a left ideal if R I \subseteq I. Similarly, a right ideal is a subset ''I'' such that I R \subseteq I. A subset ''I'' is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of ''R''. If ''E'' is a subset of ''R'', then R E is a left ideal, called the left ideal generated by ''E''; it is the smallest left ideal containing ''E''. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of ''R''. If ''x'' is in ''R'', then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by ''x''. The principal ideal RxR is written as (x). For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A c ...
of left ideals is called a left
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...
. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the
Hopkins–Levitzki theorem In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring ''R'' (with 1) is called semiprimar ...
). The integers, however, form a Noetherian ring which is not Artinian. For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal ''P'' of ''R'' is called a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
if for any elements x, y\in R we have that xy \in P implies either x \in P or y\in P. Equivalently, ''P'' is prime if for any ideals I, J we have that IJ \subseteq P implies either I \subseteq P or J \subseteq P. This latter formulation illustrates the idea of ideals as generalizations of elements.


Homomorphism

A
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from a ring to a ring is a function ''f'' from ''R'' to ''S'' that preserves the ring operations; namely, such that, for all ''a'', ''b'' in ''R'' the following identities hold: * ''f''(''a'' + ''b'') = ''f''(''a'') ‡ ''f''(''b'') * ''f''(''a'' ⋅ ''b'') = ''f''(''a'') ∗ ''f''(''b'') * ''f''(1''R'') = 1''S'' If one is working with rngs, then the third condition is dropped. A ring homomorphism ''f'' is said to be an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
if there exists an inverse homomorphism to ''f'' (that is, a ring homomorphism that is an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
). Any
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
ring homomorphism is a ring isomorphism. Two rings R, S are said to be isomorphic if there is an isomorphism between them and in that case one writes R \simeq S. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism. Examples: * The function that maps each integer ''x'' to its remainder modulo 4 (a number in ) is a homomorphism from the ring Z to the quotient ring Z/4Z ("quotient ring" is defined below). * If u is a unit element in a ring ''R'', then R \to R, x \mapsto uxu^ is a ring homomorphism, called an
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
of ''R''. * Let ''R'' be a commutative ring of prime characteristic ''p''. Then x \mapsto x^p is a ring endomorphism of ''R'' called the
Frobenius homomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
. * The
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of a field extension L/K is the set of all automorphisms of ''L'' whose restrictions to ''K'' are the identity. * For any ring ''R'', there are a unique ring homomorphism and a unique ring homomorphism . * An
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
(that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map is an epimorphism. * An algebra homomorphism from a ''k''-algebra to the
endomorphism algebra In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gro ...
of a vector space over ''k'' is called a representation of the algebra. Given a ring homomorphism f:R \to S, the set of all elements mapped to 0 by ''f'' is called the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of ''f''. The kernel is a two-sided ideal of ''R''. The image of ''f'', on the other hand, is not always an ideal, but it is always a subring of ''S''. To give a ring homomorphism from a commutative ring ''R'' to a ring ''A'' with image contained in the center of ''A'' is the same as to give a structure of an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
over ''R'' to ''A'' (which in particular gives a structure of an ''A''-module).


Quotient ring

The notion of
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
is analogous to the notion of a
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
. Given a ring and a two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
''I'' of , view ''I'' as subgroup of ; then the quotient ring ''R''/''I'' is the set of
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of ''I'' together with the operations :(''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I'' and :(''a'' + ''I'')(''b'' + ''I'') = (''ab'') + ''I''. for all ''a'', ''b'' in ''R''. The ring ''R''/''I'' is also called a factor ring. As with a quotient group, there is a canonical homomorphism p \colon R \to R/I, given by x \mapsto x + I. It is surjective and satisfies the following universal property: *If f \colon R \to S is a ring homomorphism such that f(I) = 0, then there is a unique homomorphism \overline \colon R/I \to S such that f = \overline \circ p. For any ring homomorphism f \colon R \to S, invoking the universal property with I = \ker f produces a homomorphism \overline \colon R/\ker f \to S that gives an isomorphism from R/\ker f to the image of .


Module

The concept of a ''module over a ring'' generalizes the concept of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
(over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
) by generalizing from multiplication of vectors with elements of a field (
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
) to multiplication with elements of a ring. More precisely, given a ring with 1, an -module is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
equipped with an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
(associating an element of to every pair of an element of and an element of ) that satisfies certain
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all in and all in , we have: * is an abelian group under addition. * a(x+y)=ax+ay * (a+b)x=ax+bx * 1x=x * (ab)x=a(bx) When the ring is
noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
these axioms define ''left modules''; ''right modules'' are defined similarly by writing instead of . This is not only a change of notation, as the last axiom of right modules (that is ) becomes , if left multiplication (by ring elements) is used for a right module. Basic examples of modules are ideals, including the ring itself. Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the
dimension of a vector space In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to di ...
). In particular, not all modules have a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
. The axioms of modules imply that , where the first minus denotes the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers. Any ring homomorphism induces a structure of a module: if is a ring homomorphism, then is a left module over by the multiplication: . If is commutative or if is contained in the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of , the ring is called a -
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
. In particular, every ring is an algebra over the integers.


Constructions


Direct product

Let ''R'' and ''S'' be rings. Then the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
can be equipped with the following natural ring structure: * (''r''1, ''s''1) + (''r''2, ''s''2) = (''r''1 + ''r''2, ''s''1 + ''s''2) * (''r''1, ''s''1) ⋅ (''r''2, ''s''2) = (''r''1 ⋅ ''r''2, ''s''1 ⋅ ''s''2) for all ''r''1, ''r''2 in ''R'' and ''s''1, ''s''2 in ''S''. The ring with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of ''R'' with ''S''. The same construction also works for an arbitrary family of rings: if R_i are rings indexed by a set ''I'', then \prod_ R_i is a ring with componentwise addition and multiplication. Let ''R'' be a commutative ring and \mathfrak_1, \cdots, \mathfrak_n be ideals such that \mathfrak_i + \mathfrak_j = (1) whenever i \ne j. Then the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
says there is a canonical ring isomorphism: R / \simeq \prod_^, \qquad x \bmod \mapsto (x \bmod \mathfrak_1, \ldots , x \bmod \mathfrak_n). A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let R_i, 1 \le i \le n be rings, R_i \to R = \prod R_i the inclusions with the images \mathfrak_i (in particular \mathfrak_i are rings though not subrings). Then \mathfrak_i are ideals of ''R'' and R = \mathfrak_1 \oplus \cdots \oplus \mathfrak_n, \quad \mathfrak_i \mathfrak_j = 0, i \ne j, \quad \mathfrak_i^2 \subseteq \mathfrak_i as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to ''R''. Equivalently, the above can be done through
central idempotent In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element ''a'' such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for ...
s. Assume that ''R'' has the above decomposition. Then we can write 1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak_i. By the conditions on \mathfrak_i, one has that e_i are central idempotents and e_i e_j = 0, i \ne j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let \mathfrak_i = R e_i, which are two-sided ideals. If each e_i is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to ''R''. An important application of an infinite direct product is the construction of a
projective limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits c ...
of rings (see below). Another application is a
restricted product In mathematics, the restricted product is a construction in the theory of topological groups. Let I be an index set; S a finite subset of I. If G_i is a locally compact group for each i \in I, and K_i \subset G_i is an open compact subgroup for e ...
of a family of rings (cf.
adele ring Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
).


Polynomial ring

Given a symbol ''t'' (called a variable) and a commutative ring ''R'', the set of polynomials : R = \left\ forms a commutative ring with the usual addition and multiplication, containing ''R'' as a subring. It is called the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
over ''R''. More generally, the set R\left _1, \ldots, t_n\right/math> of all polynomials in variables t_1, \ldots, t_n forms a commutative ring, containing R\left _i\right/math> as subrings. If ''R'' is an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
, then R /math> is also an integral domain; its field of fractions is the field of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s. If ''R'' is a Noetherian ring, then R /math> is a Noetherian ring. If ''R'' is a unique factorization domain, then R /math> is a unique factorization domain. Finally, ''R'' is a field if and only if R /math> is a principal ideal domain. Let R \subseteq S be commutative rings. Given an element ''x'' of ''S'', one can consider the ring homomorphism : R \to S, \quad f \mapsto f(x) (that is, the substitution). If and , then . Because of this, the polynomial ''f'' is often also denoted by f(t). The image of the map f \mapsto f(x) is denoted by R /math>; it is the same thing as the subring of ''S'' generated by ''R'' and ''x''. Example: k\left ^2, t^3\right/math> denotes the image of the homomorphism :k
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
\to k \, f \mapsto f\left(t^2, t^3\right). In other words, it is the subalgebra of k /math> generated by ''t''2 and ''t''3. Example: let ''f'' be a polynomial in one variable, that is, an element in a polynomial ring ''R''. Then f(x+h) is an element in R /math> and f(x + h) - f(x) is divisible by ''h'' in that ring. The result of substituting zero to ''h'' in (f(x + h) - f(x))/h is f'(x), the derivative of ''f'' at ''x''. The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism \phi: R \to S and an element ''x'' in ''S'' there exists a unique ring homomorphism \overline: R \to S such that \overline(t) = x and \overline restricts to \phi. For example, choosing a basis, a
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
satisfies the universal property and so is a polynomial ring. To give an example, let ''S'' be the ring of all functions from ''R'' to itself; the addition and the multiplication are those of functions. Let ''x'' be the identity function. Each ''r'' in ''R'' defines a constant function, giving rise to the homomorphism R \to S. The universal property says that this map extends uniquely to :R \to S, \quad f \mapsto \overline (''t'' maps to ''x'') where \overline is the
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
defined by ''f''. The resulting map is injective if and only if ''R'' is infinite. Given a non-constant monic polynomial ''f'' in R /math>, there exists a ring ''S'' containing ''R'' such that ''f'' is a product of linear factors in S /math>. Let ''k'' be an algebraically closed field. The
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
(theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in k\left _1, \ldots, t_n\right/math> and the set of closed subvarieties of k^n. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf.
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbn ...
.) There are some other related constructions. A
formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
R ![t!.html"_;"title=".html"_;"title="![t">![t!">.html"_;"title="![t">![t!/math>_consists_of_formal_power_series :_\sum_0^\infty_a_i_t^i,_\quad_a_i_\in_R together_with_multiplication_and_addition_that_mimic_those_for_convergent_series._It_contains_R_/math>_as_a_subring._A_formal_power_series_ring_does_not_have_the_universal_property_of_a_polynomial_ring;_a_series_may_not_converge_after_a_substitution._The_important_advantage_of_a_formal_power_series_ring_over_a_polynomial_ring_is_that_it_is_
![t!.html"_;"title=".html"_;"title="![t">![t!">.html"_;"title="![t">![t!/math>_consists_of_formal_power_series :_\sum_0^\infty_a_i_t^i,_\quad_a_i_\in_R together_with_multiplication_and_addition_that_mimic_those_for_convergent_series._It_contains_R_/math>_as_a_subring._A_formal_power_series_ring_does_not_have_the_universal_property_of_a_polynomial_ring;_a_series_may_not_converge_after_a_substitution._The_important_advantage_of_a_formal_power_series_ring_over_a_polynomial_ring_is_that_it_is_local_ring">local_ Local_may_refer_to: _Geography_and_transportation *_Local_(train),_a_train_serving_local_traffic_demand *_Local,_Missouri,_a_community_in_the_United_States *_Local_government,_a_form_of_public_administration,_usually_the_lowest_tier_of_administrat_...
_(in_fact,_complete_ring.html" "title="local_ring.html" "title="">![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math> consists of formal power series : \sum_0^\infty a_i t^i, \quad a_i \in R together with multiplication and addition that mimic those for convergent series. It contains R /math> as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local ring">local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
(in fact, complete ring">complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
).


Matrix ring and endomorphism ring

Let ''R'' be a ring (not necessarily commutative). The set of all square matrices of size ''n'' with entries in ''R'' forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by M''n''(''R''). Given a right ''R''-module U, the set of all ''R''-linear maps from ''U'' to itself forms a ring with addition that is of function and multiplication that is of
composition of functions In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
; it is called the endomorphism ring of ''U'' and is denoted by \operatorname_R(U). As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: \operatorname_R(R^n) \simeq \operatorname_n(R). This is a special case of the following fact: If f: \oplus_1^n U \to \oplus_1^n U is an ''R''-linear map, then ''f'' may be written as a matrix with entries f_ in S = \operatorname_R(U), resulting in the ring isomorphism: :\operatorname_R(\oplus_1^n U) \to \operatorname_n(S), \quad f \mapsto (f_). Any ring homomorphism induces .
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
says that if ''U'' is a simple right ''R''-module, then \operatorname_R(U) is a division ring. If \textstyle U = \bigoplus_^r U_i^ is a direct sum of ''m''''i''-copies of simple ''R''-modules U_i, then :\operatorname_R(U) \simeq \prod_^r \operatorname_ (\operatorname_R(U_i)). The Artin–Wedderburn theorem states any
semisimple ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
(cf. below) is of this form. A ring ''R'' and the matrix ring M''n''(''R'') over it are
Morita equivalent In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
: the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of right modules of ''R'' is equivalent to the category of right modules over M''n''(''R''). In particular, two-sided ideals in ''R'' correspond in one-to-one to two-sided ideals in M''n''(''R'').


Limits and colimits of rings

Let ''R''''i'' be a sequence of rings such that ''R''''i'' is a subring of ''R''''i''+1 for all ''i''. Then the union (or
filtered colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...
) of ''R''''i'' is the ring \varinjlim R_i defined as follows: it is the disjoint union of all ''R''''i'''s modulo the equivalence relation x \sim y if and only if x = y in ''R''''i'' for sufficiently large ''i''. Examples of colimits: * A polynomial ring in infinitely many variables: R _1, t_2, \cdots= \varinjlim R _1, t_2, \cdots, t_m * The
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s of the same characteristic \overline_p = \varinjlim \mathbf_. * The field of
formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
over a field ''k'': k(\!(t)\!) = \varinjlim t^k ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math> (it is the field of fractions of the
formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
k ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math>.) * The function field of an algebraic variety over a field ''k'' is \varinjlim k[U] where the limit runs over all the coordinate rings k[U] of nonempty open subsets ''U'' (more succinctly it is the stalk (mathematics), stalk of the structure sheaf at the
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic g ...
.) Any commutative ring is the colimit of finitely generated subrings. A
projective limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits c ...
(or a filtered limit) of rings is defined as follows. Suppose we're given a family of rings R_i, ''i'' running over positive integers, say, and ring homomorphisms R_j \to R_i, j \ge i such that R_i \to R_i are all the identities and R_k \to R_j \to R_i is R_k \to R_i whenever k \ge j \ge i. Then \varprojlim R_i is the subring of \textstyle \prod R_i consisting of (x_n) such that x_j maps to x_i under R_j \to R_i, j \ge i. For an example of a projective limit, see .


Localization

The
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
generalizes the construction of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring ''R'' and a subset ''S'' of ''R'', there exists a ring R ^/math> together with the ring homomorphism R \to R\left ^\right/math> that "inverts" ''S''; that is, the homomorphism maps elements in ''S'' to unit elements in R\left ^\right/math>, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through R\left ^\right/math>. The ring R\left ^\right/math> is called the localization of ''R'' with respect to ''S''. For example, if ''R'' is a commutative ring and ''f'' an element in ''R'', then the localization R\left ^\right/math> consists of elements of the form r/f^n, \, r \in R , \, n \ge 0 (to be precise, R\left ^\right= R (tf - 1).) The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in ''R''. In that case S = R - \mathfrak, one often writes R_\mathfrak for R\left ^\right/math>. R_\mathfrak is then a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
with the
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
\mathfrak R_\mathfrak. This is the reason for the terminology "localization". The field of fractions of an integral domain ''R'' is the localization of ''R'' at the prime ideal zero. If \mathfrak is a prime ideal of a commutative ring ''R'', then the field of fractions of R/\mathfrak is the same as the residue field of the local ring R_\mathfrak and is denoted by k(\mathfrak). If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a
change of rings In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operatorn ...
M\left ^\right= R\left ^\right\otimes_R M. The most important properties of localization are the following: when ''R'' is a commutative ring and ''S'' a multiplicatively closed subset * \mathfrak \mapsto \mathfrak\left ^\right/math> is a bijection between the set of all prime ideals in ''R'' disjoint from ''S'' and the set of all prime ideals in R\left ^\right/math>. * R\left ^\right= \varinjlim R\left ^\right/math>, ''f'' running over elements in ''S'' with partial ordering given by divisibility. * The localization is exact: 0 \to M'\left ^\right\to M\left ^\right\to M''\left ^\right\to 0 is exact over R\left ^\right/math> whenever 0 \to M' \to M \to M'' \to 0 is exact over ''R''. * Conversely, if 0 \to M'_\mathfrak \to M_\mathfrak \to M''_\mathfrak \to 0 is exact for any maximal ideal \mathfrak, then 0 \to M' \to M \to M'' \to 0 is exact. * A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.) In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a
localization of a category In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in gene ...
amounts to making some morphisms isomorphisms. An element in a commutative ring ''R'' may be thought of as an endomorphism of any ''R''-module. Thus, categorically, a localization of ''R'' with respect to a subset ''S'' of ''R'' is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the category of ''R''-modules to itself that sends elements of ''S'' viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, ''R'' then maps to R\left ^\right/math> and ''R''-modules map to R\left ^\right/math>-modules.)


Completion

Let ''R'' be a commutative ring, and let ''I'' be an ideal of ''R''. The completion of ''R'' at ''I'' is the projective limit \hat = \varprojlim R/I^n; it is a commutative ring. The canonical homomorphisms from ''R'' to the quotients R/I^n induce a homomorphism R \to \hat. The latter homomorphism is injective if ''R'' is a Noetherian integral domain and ''I'' is a proper ideal, or if ''R'' is a Noetherian local ring with maximal ideal ''I'', by Krull's intersection theorem. The construction is especially useful when ''I'' is a maximal ideal. The basic example is the completion of Z at the principal ideal (''p'') generated by a prime number ''p''; it is called the ring of ''p''-adic integers and is denoted Z''p''. The completion can in this case be constructed also from the ''p''-adic absolute value on Q. The ''p''-adic absolute value on Q is a map x \mapsto , x, from Q to R given by , n, _p=p^ where v_p(n) denotes the exponent of ''p'' in the prime factorization of a nonzero integer ''n'' into prime numbers (we also put , 0, _p=0 and , m/n, _p = , m, _p/, n, _p). It defines a distance function on Q and the completion of Q as a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
is denoted by Q''p''. It is again a field since the field operations extend to the completion. The subring of Q''p'' consisting of elements ''x'' with , x, _p \le 1 is isomorphic to Z''p''. Similarly, the formal power series ring R[] is the completion of R /math> at (t) (see also Hensel's lemma) A complete ring has much simpler structure than a commutative ring. This owns to the
Cohen structure theorem In mathematics, the Cohen structure theorem, introduced by , describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull: *Any complete regular equicharacteristic ...
, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of
excellent ring In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the ...
.


Rings with generators and relations

The most general way to construct a ring is by specifying generators and relations. Let ''F'' be a free ring (that is, free algebra over the integers) with the set ''X'' of symbols, that is, ''F'' consists of polynomials with integral coefficients in noncommuting variables that are elements of ''X''. A free ring satisfies the universal property: any function from the set ''X'' to a ring ''R'' factors through ''F'' so that F \to R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring. Now, we can impose relations among symbols in ''X'' by taking a quotient. Explicitly, if ''E'' is a subset of ''F'', then the quotient ring of ''F'' by the ideal generated by ''E'' is called the ring with generators ''X'' and relations ''E''. If we used a ring, say, ''A'' as a base ring instead of Z, then the resulting ring will be over ''A''. For example, if E = \, then the resulting ring will be the usual polynomial ring with coefficients in ''A'' in variables that are elements of ''X'' (It is also the same thing as the
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
over ''A'' with symbols ''X''.) In the category-theoretic terms, the formation S \mapsto \text S is the left adjoint functor of the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
from the
category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings ...
to Set (and it is often called the free ring functor.) Let ''A'', ''B'' be algebras over a commutative ring ''R''. Then the tensor product of ''R''-modules A \otimes_R B is an ''R''-algebra with multiplication characterized by (x \otimes u) (y \otimes v) = xy \otimes uv.


Special kinds of rings


Domains

A nonzero ring with no nonzero
zero-divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s is called a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
. A commutative domain is called an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
(UFD), an integral domain in which every nonunit element is a product of
prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...
s (an element is prime if it generates a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
.) The fundamental question in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
is on the extent to which the ring of (generalized) integers in a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
, where an "ideal" admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the
structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitel ...
. The theorem may be illustrated by the following application to linear algebra. Let ''V'' be a finite-dimensional vector space over a field ''k'' and f: V \to V a linear map with minimal polynomial ''q''. Then, since k /math> is a unique factorization domain, ''q'' factors into powers of distinct irreducible polynomials (that is, prime elements): q = p_1^ \ldots p_s^. Letting t \cdot v = f(v), we make ''V'' a ''k'' 't''module. The structure theorem then says ''V'' is a direct sum of
cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-mod ...
s, each of which is isomorphic to the module of the form k / \left(p_i^\right). Now, if p_i(t) = t - \lambda_i, then such a cyclic module (for p_i) has a basis in which the restriction of ''f'' is represented by a
Jordan matrix In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the ...
. Thus, if, say, ''k'' is algebraically closed, then all p_i's are of the form t - \lambda_i and the above decomposition corresponds to the
Jordan canonical form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
of ''f''. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a
regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
. A regular local ring is a UFD. The following is a chain of class inclusions that describes the relationship between rings, domains and fields:


Division ring

A
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
is a ring such that every non-zero element is a unit. A commutative division ring is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. A prominent example of a division ring that is not a field is the ring of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every ''finite'' domain (in particular finite division ring) is a field; in particular commutative (the
Wedderburn's little theorem In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to altern ...
). Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem. A
cyclic algebra In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field, and plays a key role in the theory of central simple algebras. Definition Let ''A'' be a finite-dimensional central simple algebra over a fiel ...
, introduced by
L. E. Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
, is a generalization of a
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
.


Semisimple rings

A ''
semisimple module In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
'' is a direct sum of simple modules. A ''
semisimple ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
'' is a ring that is semisimple as a left module (or right module) over itself.


Examples

* A
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
is semisimple (and
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
). * For any division ring and positive integer , the matrix ring is semisimple (and
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
). * For a field and finite group , the group ring is semisimple if and only if the characteristic of does not divide the order of (
Maschke's theorem In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
). *
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
s are semisimple. The
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
over a field is a
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ...
, but it is not semisimple. The same holds for a ring of differential operators in many variables.


Properties

Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.) For a ring , the following are equivalent: * is semisimple. * is artinian and semiprimitive. * is a finite
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
\prod_^r \operatorname_(D_i) where each is a positive integer, and each is a division ring ( Artin–Wedderburn theorem). Semisimplicity is closely related to separability. A unital associative algebra ''A'' over a field ''k'' is said to be separable if the base extension A \otimes_k F is semisimple for every
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
F/k. If ''A'' happens to be a field, then this is equivalent to the usual definition in field theory (cf.
separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
.)


Central simple algebra and Brauer group

For a field ''k'', a ''k''-algebra is central if its center is ''k'' and is simple if it is a
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ...
. Since the center of a simple ''k''-algebra is a field, any simple ''k''-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a ''k''-algebra. The matrix ring of size ''n'' over a ring ''R'' will be denoted by R_n. The
Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in ...
states any automorphism of a central simple algebra is inner. Two central simple algebras ''A'' and ''B'' are said to be ''similar'' if there are integers ''n'' and ''m'' such that A \otimes_k k_n \approx B \otimes_k k_m. Since k_n \otimes_k k_m \simeq k_, the similarity is an equivalence relation. The similarity classes /math> with the multiplication B] = \left \otimes_k B\right/math> form an abelian group called the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
of ''k'' and is denoted by \operatorname(k). By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, \operatorname(k) is trivial if ''k'' is a finite field or an algebraically closed field (more generally
quasi-algebraically closed field In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebra ...
; cf.
Tsen's theorem In mathematics, Tsen's theorem states that a function field ''K'' of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally t ...
). \operatorname(\mathbf) has order 2 (a special case of the theorem of Frobenius). Finally, if ''k'' is a nonarchimedean
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
(for example, \mathbf_p), then \operatorname(k) = \mathbf/\mathbf through the invariant map. Now, if ''F'' is a field extension of ''k'', then the base extension - \otimes_k F induces \operatorname(k) \to \operatorname(F). Its kernel is denoted by \operatorname(F/k). It consists of /math> such that A \otimes_k F is a matrix ring over ''F'' (that is, ''A'' is split by ''F''.) If the extension is finite and Galois, then \operatorname(F/k) is canonically isomorphic to H^2\left(\operatorname(F/k), k^*\right).
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local rin ...
s generalize the notion of central simple algebras to a commutative local ring.


Valuation ring

If is a field, a valuation is a group homomorphism from the multiplicative group to a totally ordered abelian group such that, for any in with nonzero, . The
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such t ...
of is the subring of consisting of zero and all nonzero such that . Examples: * The field of
formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
k(\!(t)\!) over a field comes with the valuation such that is the least degree of a nonzero term in ; the valuation ring of is the
formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
k ![t!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math>. * More generally, given a field and a totally ordered abelian group , let k(\!(G)\!) be the set of all functions from to whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
: (f*g)(t) = \sum_ f(s)g(t - s). It also comes with the valuation such that is the least element in the support of . The subring consisting of elements with finite support is called the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
of (which makes sense even if is not commutative). If is the ring of integers, then we recover the previous example (by identifying with the series whose -th coefficient is .)


Rings with extra structure

A ring may be viewed as an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
(by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example: * An
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
is a ring that is also a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a field ''K'' such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of ''n''-by-''n'' matrices over the real field R has dimension ''n''2 as a real vector space. * A ring ''R'' is a
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...
if its set of elements ''R'' is given a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
which makes the addition map ( + : R\times R \to R\,) and the multiplication map ( \cdot : R\times R \to R\,) to be both
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
as maps between topological spaces (where ''X'' × ''X'' inherits the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
, or the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, and in either case one would obtain a topological ring. * A
λ-ring In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide ...
is a commutative ring ''R'' together with operations that are like ''n''-th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
s: ::\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^(y). :For example, Z is a λ-ring with \lambda^n(x) = \binom, the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. The notion plays a central rule in the algebraic approach to the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
. * A
totally ordered ring In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'': * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. * if 0 ≤ ''a'' and 0 ≤ ''b'' then ...
is a ring with a
total ordering In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
that is compatible with ring operations.


Some examples of the ubiquity of rings

Many different kinds of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical pr ...
s can be fruitfully analyzed in terms of some associated ring.


Cohomology ring of a topological space

To any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' one can associate its integral
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
:H^*(X,\mathbf) = \bigoplus_^ H^i(X,\mathbf), a
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
. There are also
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s H_i(X,\mathbf) of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s and tori, for which the methods of
point-set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
are not well-suited.
Cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...
. However, the advantage of the cohomology groups is that there is a
natural product A natural product is a natural compound or substance produced by a living organism—that is, found in nature. In the broadest sense, natural products include any substance produced by life. Natural products can also be prepared by chemical syn ...
, which is analogous to the observation that one can multiply pointwise a ''k''-
multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
and an ''l''-multilinear form to get a ()-multilinear form. The ring structure in cohomology provides the foundation for
characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es of
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s, intersection theory on manifolds and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
,
Schubert calculus In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ...
and much more.


Burnside ring of a group

To any
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
is associated its
Burnside ring In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic r ...
which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the
representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representati ...
: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.


Representation ring of a group ring

To any
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
or Hopf algebra is associated its
representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representati ...
or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from
character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about ...
, which is more or less the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
given a ring structure.


Function field of an irreducible algebraic variety

To any irreducible
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
is associated its function field. The points of an algebraic variety correspond to
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such t ...
s contained in the function field and containing the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
. The study of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
makes heavy use of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
to study geometric concepts in terms of ring-theoretic properties.
Birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
studies maps between the subrings of the function field.


Face ring of a simplicial complex

Every
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
has an associated face ring, also called its
Stanley–Reisner ring In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ...
. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in
algebraic combinatorics Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb ...
. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of
simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...
s.


Category-theoretic description

Every ring can be thought of as a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
in Ab, the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...
(thought of as a
monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
under the tensor product of -modules). The monoid action of a ring ''R'' on an abelian group is simply an ''R''-module. Essentially, an ''R''-module is a generalization of the notion of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
– where rather than a vector space over a field, one has a "vector space over a ring". Let be an abelian group and let End(''A'') be its
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
(see above). Note that, essentially, End(''A'') is the set of all morphisms of ''A'', where if ''f'' is in End(''A''), and ''g'' is in End(''A''), the following rules may be used to compute and : * (''f'' + ''g'')(''x'') = ''f''(''x'') + ''g''(''x'') * (''f'' ⋅ ''g'')(''x'') = ''f''(''g''(''x'')), where + as in is addition in ''A'', and function composition is denoted from right to left. Therefore,
associated Associated may refer to: *Associated, former name of Avon, Contra Costa County, California * Associated Hebrew Schools of Toronto, a school in Canada *Associated Newspapers, former name of DMG Media, a British publishing company See also *Associati ...
to any abelian group, is a ring. Conversely, given any ring, , is an abelian group. Furthermore, for every ''r'' in ''R'', right (or left) multiplication by ''r'' gives rise to a morphism of , by right (or left) distributivity. Let . Consider those
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s of ''A'', that "factor through" right (or left) multiplication of ''R''. In other words, let End''R''(''A'') be the set of all morphisms ''m'' of ''A'', having the property that . It was seen that every ''r'' in ''R'' gives rise to a morphism of ''A'': right multiplication by ''r''. It is in fact true that this association of any element of ''R'', to a morphism of ''A'', as a function from ''R'' to End''R''(''A''), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian ''X''-group (by ''X''-group, it is meant a group with ''X'' being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian ''X''-group. Any ring can be seen as a
preadditive category In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom ...
with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context.
Additive functor In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom- ...
s between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s closed under addition and under composition with arbitrary morphisms.


Generalization

Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.


Rng

A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.


Nonassociative ring

A
nonassociative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' i ...
is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.


Semiring

A
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
(sometimes ''rig'') is obtained by weakening the assumption that (''R'', +) is an abelian group to the assumption that (''R'', +) is a commutative monoid, and adding the axiom that for all ''a'' in ''R'' (since it no longer follows from the other axioms). Examples: * the non-negative integers \ with ordinary addition and multiplication; * the
tropical semiring In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical s ...
.


Other ring-like objects


Ring object in a category

Let ''C'' be a category with finite
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
. Let pt denote a
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
of ''C'' (an empty product). A ring object in ''C'' is an object ''R'' equipped with morphisms R \times R\;\stackrel\to\,R (addition), R \times R\;\stackrel\to\,R (multiplication), \operatorname\stackrel\to\,R (additive identity), R\;\stackrel\to\,R (additive inverse), and \operatorname\stackrel\to\,R (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object ''R'' equipped with a factorization of its functor of points h_R = \operatorname(-,R) : C^ \to \mathbf through the category of rings: C^ \to \mathbf \stackrel\longrightarrow \mathbf.


Ring scheme

In algebraic geometry, a ring scheme over a base
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
is a ring object in the category of -schemes. One example is the ring scheme over , which for any commutative ring returns the ring of -isotypic
Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of ord ...
s of length over .Serre, p. 44.


Ring spectrum

In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a
ring spectrum In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy a ...
is a
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
''X'' together with a multiplication \mu \colon X \wedge X \to X and a unit map S \to X from the sphere spectrum ''S'', such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a
monoid object In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η' ...
in a good category of spectra such as the category of
symmetric spectra In algebraic topology, a symmetric spectrum ''X'' is a spectrum of pointed simplicial sets that comes with an action of the symmetric group \Sigma_n on X_n such that the composition of structure maps :S^1 \wedge \dots \wedge S^1 \wedge X_n \to S^1 \ ...
.


See also

*
Algebra over a commutative ring In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
*
Categorical ring In mathematics, a categorical ring is, roughly, a Category (mathematics), category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a Ring (mathematics), ring by a category ...
*
Category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings ...
*
Glossary of ring theory Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory ...
*
Nonassociative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' i ...
*
Ring of sets In mathematics, there are two different notions of a ring of sets, both referring to certain Family of sets, families of sets. In order theory, a nonempty family of sets \mathcal is called a ring (of sets) if it is closure (mathematics), closed u ...
*
Semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
*
Spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
*
Simplicial commutative ring In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be sho ...
Special types of rings: *
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
*
Dedekind ring In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
*
Differential ring In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
* Exponential ring *
Finite ring In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite grou ...
*
Lie ring In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
*
Local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
*
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
and
artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...
s *
Ordered ring In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'': * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. * if 0 ≤ ''a'' and 0 ≤ ''b'' then ...
* Poisson ring *
Reduced ring In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' =&n ...
*
Regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
*
Ring of periods In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods Closure (mathematics), remain periods, so the periods form a ring (mathematics), r ...
*
SBI ring In algebra, an SBI ring is a ring ''R'' (with identity) such that every idempotent of ''R'' modulo the Jacobson radical can be lifted to ''R''. The abbreviation SBI was introduced by Irving Kaplansky and stands for "suitable for building idempoten ...
*
Valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such t ...
and
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...


Notes


Citations


References


General references

* * * * * . * * * * * * * * * . * * * * . * * * . * . * * *


Special references

* * * * * * * * * * * * * * * * * * * *


Primary sources

* * *


Historical references


History of ring theory at the MacTutor Archive
*
Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Geo ...
and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
(1996) ''A Survey of Modern Algebra'', 5th ed. New York: Macmillan. * Bronshtein, I. N. and Semendyayev, K. A. (2004)
Handbook of Mathematics ''Bronshtein and Semendyayev'' (often just ''Bronshtein'' or ''Bronstein'', sometimes ''BS'') is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Rus ...
, 4th ed. New York: Springer-Verlag . * Faith, Carl (1999) ''Rings and things and a fine array of twentieth century associative algebra''. Mathematical Surveys and Monographs, 65.
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. * Itô, K. editor (1986) "Rings." §368 in ''Encyclopedic Dictionary of Mathematics'', 2nd ed., Vol. 2. Cambridge, MA:
MIT Press The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States). It was established in 1962. History The MIT Press traces its origins back to 1926 when MIT publish ...
. * Israel Kleiner (1996) "The Genesis of the Abstract Ring Concept",
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
103: 417–424 * Kleiner, I. (1998) "From numbers to rings: the early history of ring theory",
Elemente der Mathematik ''Elemente der Mathematik'' is a peer-reviewed scientific journal covering mathematics. It is published by the European Mathematical Society, European Mathematical Society Publishing House on behalf of the Swiss Mathematical Society. It was establ ...
53: 18–35. *
B. L. van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amsterd ...
(1985) ''A History of Algebra'', Springer-Verlag, {{DEFAULTSORT:Ring (Mathematics) Algebraic structures Ring theory