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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 structures that generalize fields: multiplication need not be commutative 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
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 operations satisfying properties analogous to those of addition and multiplication 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 languag ...
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 languag ...
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 fo ...
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 exampl ...
s, square matrices, 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 con ...
. 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 comm ...
whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (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. Prom ...
, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory 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. 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 of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of real square matrices with , group rings in representation theory, operator algebras 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, norm, topology, etc.) and the linear functions defi ...
, rings of differential operators, and cohomology rings in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains 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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, 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. 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 (3 ...
.


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 comm ...
under addition, meaning that: #* (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, + is associative). #* ''a'' + ''b'' = ''b'' + ''a'' for all ''a'', ''b'' in ''R''   (that is, + is commutative). #* There is an element 0 in ''R'' such that ''a'' + 0 = ''a'' for all ''a'' in ''R''   (that is, 0 is the additive identity). #* For each ''a'' in ''R'' there exists −''a'' in ''R'' such that ''a'' + (−''a'') = 0   (that is, −''a'' is the additive inverse 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. Monoid ...
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). # 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 integers 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, 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 rings''. 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 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
is called a field. 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 ...
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 number ...
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 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. * If a ring ''R'' contains the zero ring as a subring, then ''R'' itself is the zero ring. * The binomial formula 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 with entries in a field is :\operatorname_2(F) = \left\. With the operations of matrix addition and matrix multiplication, \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.


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 integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer'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 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 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
. In 1921, Emmy Noether 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 t ...
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. * A unital associative algebra over a commutative ring 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 s ...
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 g ...
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, the integral closure of \mathbf in a quadratic extension of \mathbf. It is a subring of the ring of all algebraic integers. * The ring of profinite integers \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 of becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.


Noncommutative rings

* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n'' matrices 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 comm ...
, then the endomorphisms of ''G'' form a ring, the endomorphism ring 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. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero. * If ''G'' is a group and ''R'' is a ring, the group ring of ''G'' over ''R'' is a free module 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 (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras 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 ...
s . with the usual operations is not a ring, since is not even a group (the elements are not all invertible 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 (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 mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
: (f * g)(x) = \int_^\infty f(y)g(x - y) \, dy. Then ''R'' is a rng, but not a ring: the Dirac delta function 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 ze ...
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 is an element a such that a^n = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor. An idempotent e is an element such that e^2 = e. One example of an idempotent element is a projection in linear algebra. A unit 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
; in this case the inverse is unique, and is denoted by a^. The set of units of a ring is a group 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.


Subring

A subset ''S'' of ''R'' is called a subring if any one of the following equivalent conditions holds: * the addition and multiplication of ''R'' restrict 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 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 exampl ...
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 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 (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 of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. 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 semiprimary ...
). 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 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 morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 i ...
if there exists an inverse homomorphism to ''f'' (that is, a ring homomorphism that is an inverse function). Any bijective 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 itse ...
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. * The Galois group 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 (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 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 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 ...
over ''R'' to ''A'' (which in particular gives a structure of an ''A''-module).


Quotient ring

The notion of quotient ring is analogous to the notion of a quotient group. 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 considered ...
''I'' of , view ''I'' as subgroup of ; then the quotient ring ''R''/''I'' is the set of cosets 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) by generalizing from multiplication of vectors with elements of a field ( scalar multiplication) 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 comm ...
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-Man ...
(associating an element of to every pair of an element of and an element of ) that satisfies certain axioms. 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 particular, not all modules have a basis. The axioms of modules imply that , where the first minus denotes the additive inverse 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 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 ...
. In particular, every ring is an algebra over the integers.


Constructions


Direct product

Let ''R'' and ''S'' be rings. Then the product 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 t ...
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 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 fo ...
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 ca ...
of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).


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, 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 ...
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 Substitution may refer to: Arts and media *Chord substitution, in music, swapping one chord for a related one within a chord progression *Substitution (poetry), a variation in poetic scansion * "Substitution" (song), a 2009 song by Silversun Pic ...
). 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\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 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 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 (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.) There are some other related constructions. A formal power series ring 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_(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 (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; 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 (cf. below) is of this form. A ring ''R'' and the matrix ring M''n''(''R'') over it are Morita equivalent: the category 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) 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 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 su ...
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 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.) 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 ca ...
(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 generalizes the construction of the field of fractions 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 algebrai ...
with the maximal ideal \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 = \operator ...
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, ca ...
, 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 gen ...
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 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 setti ...
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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language ...
, 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 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.


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 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 from 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 * ...
. A commutative domain is called an integral domain. 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 (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, 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 finite ...
. 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 modules, 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. 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 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. 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 is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. 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). 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 In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings such that ''xKx''−1 is contained in ''K'' ...
. A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.


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'' is a ring that is semisimple as a left module (or right module) over itself.


Examples

* A division 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 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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
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 algebras are semisimple. The Weyl algebra 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 simpl ...
, 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 t ...
\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 F/k. If ''A'' happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)


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 simpl ...
. 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 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; cf. Tsen's theorem). \operatorname(\mathbf) has order 2 (a special case of the theorem of Frobenius). Finally, if ''k'' is a nonarchimedean local field (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 algebras 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 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 su ...
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 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 mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
: (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 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 comm ...
(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 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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 g ...
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-seem ...
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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
, or the Zariski topology, and in either case one would obtain a topological ring. * A λ-ring 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 ...
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. * A totally ordered ring is a ring with a total ordering that is compatible with ring operations.


Some examples of the ubiquity of rings

Many different kinds of mathematical objects 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 poin ...
''X'' one can associate its integral cohomology ring :H^*(X,\mathbf) = \bigoplus_^ H^i(X,\mathbf), a graded ring. There are also homology groups 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 geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
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 groups 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. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a ''k''- multilinear form and an ''l''-multilinear form to get a ()-multilinear form. The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.


Burnside ring of a group

To any group is associated its Burnside ring 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 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: 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 or Hopf algebra is associated its representation ring 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, which is more or less the Grothendieck group 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. ...
is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. 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. Prom ...
to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.


Face ring of a simplicial complex

Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. 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.


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. Monoid ...
in Ab, the category of abelian groups (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 ...
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 (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 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 endomorphisms 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 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 functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms 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 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. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.


Semiring

A semiring (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.


Other ring-like objects


Ring object in a category

Let ''C'' be a category with finite products. Let pt denote a terminal object 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 vectors 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 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 ...
''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 a good category of spectra such as the category of symmetric spectra.


See also

* Algebra over a commutative ring * Categorical ring * Category of rings * Glossary of ring theory * Nonassociative ring * Ring of sets * Semiring * Spectrum of a ring * Simplicial commutative ring Special types of rings: * Boolean ring * Dedekind ring * Differential ring *
Exponential ring In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation. Definition A field is an algebraic structure composed of a set of elements, ''F'', two binary operations, ...
* Finite ring * Lie ring *
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 algebrai ...
* Noetherian and artinian rings * Ordered ring * Poisson ring * Reduced ring * Regular ring * Ring of periods * SBI ring * Valuation ring and discrete valuation ring


Notes


Citations


References


General references

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


Special references

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


Primary sources

* * *


Historical references


History of ring theory at the MacTutor Archive
* Garrett Birkhoff and Saunders Mac Lane (1996) ''A Survey of Modern Algebra'', 5th ed. New York: Macmillan. * Bronshtein, I. N. and Semendyayev, K. A. (2004) Handbook of Mathematics, 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, meeting ...
. * 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 publ ...
. * Israel Kleiner (1996) "The Genesis of the Abstract Ring Concept", American Mathematical Monthly 103: 417–424 * Kleiner, I. (1998) "From numbers to rings: the early history of ring theory", Elemente der Mathematik 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 Amst ...
(1985) ''A History of Algebra'', Springer-Verlag, {{DEFAULTSORT:Ring (Mathematics) Algebraic structures Ring theory