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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, rings are
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s that generalize
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
: multiplication need not be
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
and
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

s need not exist. In other words, a ''ring'' is a set equipped with two
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s satisfying properties analogous to those of
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contr ...

of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s. Ring elements may be numbers such as
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, but they may also be non-numerical objects such as
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s,
square matrices In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, and
power series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Formally, a ''ring'' is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, is distributive over the addition operation, and has a multiplicative
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. (Some authors use the term "ring" 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 is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, the theory of
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
s, is a major branch of
ring theory In algebra, ring theory is the study of ring (mathematics), 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 ...
. Its development has been greatly influenced by problems and ideas of
algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
and
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. The simplest commutative rings are those that admit division by non-zero elements; such rings are called
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. 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 geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
of an
affine algebraic variety Affine (pronounced /əˈfaɪn/) relates to 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 combination, a certai ...
, and the
ring of integersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of a number field. Examples of noncommutative rings include the ring of real
square matrices In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with ,
group ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
s in
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
,
operator algebra In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
s in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
, rings of differential operators, and
cohomology ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring (mathematics), ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomol ...
s in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind,
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
, Fraenkel, and . 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...
s that occur in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, and of
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and rings of invariants that occur in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

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 descr ...
. They later proved useful in other branches of mathematics such as
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
.

# Definition

A ring is a set ''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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
under addition, meaning that: #* (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') for all ''a'', ''b'', ''c'' in ''R''   (that is, + is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). #* ''a'' + ''b'' = ''b'' + ''a'' for all ''a'', ''b'' in ''R''   (that is, + is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

). #* 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
). #* 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 A number is a 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 ...
of ''a''). # ''R'' is a
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
). # 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, 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 , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
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 A number is a 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 ...
commutative ring in which every nonzero element has a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

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 grassl ...
. 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 inferrable 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 (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
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 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 deduct ...

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 elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
. * 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 exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 grassl ...
is :$\operatorname_2\left(F\right) = \left\.$ With the operations of matrix addition and
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, $\operatorname_2\left(F\right)$ satisfies the above ring axioms. The element $\left\left( \begin 1 & 0 \\ 0 & 1 \end\right\right)$ is the multiplicative identity of the ring. If $A = \left\left( \begin 0 & 1 \\ 1 & 0 \end \right\right)$ and $B = \left\left( \begin 0 & 1 \\ 0 & 0 \end \right\right)$, then $AB = \left\left( \begin 0 & 0 \\ 0 & 1 \end \right\right)$ while $BA = \left\left( \begin 1 & 0 \\ 0 & 0 \end \right\right)$; 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
.

# History

## Dedekind

The study of rings originated from the theory of
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and the theory of
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s.The development of Ring Theory
/ref> In 1871,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...
'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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit The album-equivalent unit is a measurement unit in music industry to define the consumption of music that equals the purchase of one album copy. This consumpti ...
). 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 Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. In 1921,
Emmy Noether Amalie Emmy Noether Emmy 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 Noeth ...

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." makes the counterargument 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
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. * A unital associative
algebra over a commutative ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is itself a ring as well as an -module. Some examples: ** The algebra of
polynomials In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with coefficients in . As an -module, is
free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decis ...

of infinite rank. ** The algebra of
formal power series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 ga ...
real-valued
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
defined on the real line forms a commutative -algebra. The operations are
pointwiseIn 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 o ...
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 . *
decimal fraction The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
s. Not free as a -module. *
square-free{{no footnotes, date=December 2015 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...
integer of the form , with . A free -module of rank 2. See ''
Quadratic integer In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are ...
''. *
Gaussian integer In , a Gaussian integer is a whose real and imaginary parts are both s. The Gaussian integers, with ordinary and of s, form an , usually written as . This integral domain is a particular case of a of s. It does not have a ing that respects ari ...
s. *
Eisenstein integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. * The previous two examples are the cases and of the cyclotomic ring . * The previous four examples are cases of the
ring of integersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of a
number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, defined as the set of
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s in . * The set of all algebraic integers in forms a ring called the
integral closureIn commutative algebra, an element ''b'' of a commutative ring In ring theory, a branch of abstract algebra, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is ca ...
of in . * If is a set, then the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of becomes a ring if we define addition to be the
symmetric difference In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of sets and multiplication to be
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
. This is an example of a
Boolean ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

## Noncommutative rings

* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n''
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, 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 group ...

of ''G'' form a ring, the
endomorphism ringIn abstract algebra, the endomorphisms of an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
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 one of the fundamental algebraic structures used in abstract algebra. A module over a ring (mathematics), ring is a generalization of the notion of vector space over a Field (mathematics), field, wherein the correspond ...
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''). * If ''G'' is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
and ''R'' is a ring, the
group ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
of ''G'' over ''R'' is a
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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''. * Many rings that appear in analysis are noncommutative. For example, most
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
s are noncommutative.

# Basic concepts

## Products and powers

For each nonnegative integer , given a sequence $\left(a_1,\ldots,a_n\right)$ 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is an element $a$ such that $a^n = 0$ for some $n > 0$. One example of a nilpotent element is a
nilpotent matrixIn linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written w ...
. A nilpotent element in a nonzero ring is necessarily a zero divisor. An
idempotent Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
$e$ is an element such that $e^2 = e$. One example of an idempotent element is a projection 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 (album), ...
is an element $a$ having a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

; 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 A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
under ring multiplication; this group is denoted by $R^\times$ or $R^*$ or $U\left(R\right)$. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
.

## Subring

A subset ''S'' of ''R'' is called a
subring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
if any one of the following equivalent conditions holds: * the addition and multiplication of ''R''
restrict In the C (programming language), C programming language, restrict is a Keyword (computer programming), keyword that can be used in Pointer (computer programming), pointer declarations. By adding this type qualifier, a programmer hints to the compi ...
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 grassl ...
of real numbers and also a subring of the ring of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 rng (algebra), 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 (algebra), 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 $\operatorname\left(R\right)$ 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 $\operatorname\left(R\right)$ is a subring of ''R'', called the Center (ring theory), 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 (ring theory), 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 ideal, principal left ideals and right ideals generated by ''x''. The principal ideal $RxR$ is written as $\left(x\right)$. 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 ring, simple 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 total order#Chains, 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). 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 ring homomorphism, homomorphism 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 if there exists an inverse homomorphism to ''f'' (that is, a ring homomorphism that is an inverse function). Any bijection, 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 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 algebra representation, 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 a ring homomorphism, 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 associative algebra, algebra 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 (ring theory), ideal ''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 $f\left(I\right) = 0$, 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 (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 grassl ...
) 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
equipped with an operation (mathematics), operation (associating an element of to every pair of an element of and an element of ) that satisfies certain axiom#Non-logical axioms, 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. * $a\left(x+y\right)=ax+ay$ * $\left(a+b\right)x=ax+bx$ * $1x=x$ * $\left(ab\right)x=a\left(bx\right)$ When the ring is noncommutative ring, noncommutative 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 (vector space), dimension of a vector space). In particular, not all modules have a basis (linear algebra), basis. The axioms of modules imply that , where the first minus denotes the
additive inverse In mathematics, the additive inverse of a number A number is a 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 ...
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 a ring, center of , the ring is called a -algebra over a ring, algebra. In particular, every ring is an algebra over the integers.

# Constructions

## Direct product

Let ''R'' and ''S'' be rings. Then the cartesian product, 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 $\left(1, 1\right)$ is called the Direct product of rings, direct product 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 $\textstyle \prod_ R_i$ is a ring with componentwise addition and multiplication. Let ''R'' be a commutative ring and $\mathfrak_1, \cdots, \mathfrak_n$ be ideals such that $\mathfrak_i + \mathfrak_j = \left(1\right)$ whenever $i \ne j$. Then the Chinese remainder theorem says there is a canonical ring isomorphism: :$R / \simeq \prod_^, \qquad x \;\operatorname\; \mapsto \left(x \;\operatorname\; \mathfrak_1, \ldots , x \;\operatorname\; \mathfrak_n\right)$. 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, $\textstyle R_i \to R = \prod R_i$ 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 idempotents. 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 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[t\right] = \left\$ forms a commutative ring with the usual addition and multiplication, containing ''R'' as a subring. It is called the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over ''R''. More generally, the set $R\left\left[t_1, \ldots, t_n\right\right]$ of all polynomials in variables $t_1, \ldots, t_n$ forms a commutative ring, containing $R\left\left[t_i\right\right]$ as subrings. If ''R'' is an integral domain, then $R\left[t\right]$ is also an integral domain; its field of fractions is the field of rational functions. If ''R'' is a Noetherian ring, then $R\left[t\right]$ is a Noetherian ring. If ''R'' is a unique factorization domain, then $R\left[t\right]$ is a unique factorization domain. Finally, ''R'' is a field if and only if $R\left[t\right]$ is a principal ideal domain. Let $R \subseteq S$ be commutative rings. Given an element ''x'' of ''S'', one can consider the ring homomorphism : $R\left[t\right] \to S, \quad f \mapsto f\left(x\right)$ (that is, the substitution (algebra), substitution). If and , then . Because of this, the polynomial ''f'' is often also denoted by $f\left(t\right)$. The image of the map $f \mapsto f\left(x\right)$ is denoted by $R\left[x\right]$; it is the same thing as the subring of ''S'' generated by ''R'' and ''x''. Example: $k\left\left[t^2, t^3\right\right]$ denotes the image of the homomorphism :$k\left[x, y\right] \to k\left[t\right], \, f \mapsto f\left\left(t^2, t^3\right\right).$ In other words, it is the subalgebra of $k\left[t\right]$ 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 $f\left(x+h\right)$ is an element in $R\left[h\right]$ and $f\left(x + h\right) - f\left(x\right)$ is divisible by ''h'' in that ring. The result of substituting zero to ''h'' in $\left(f\left(x + h\right) - f\left(x\right)\right)/h$ is $f\text{'}\left(x\right)$, 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 $\overline: R\left[t\right] \to S$ such that $\overline\left(t\right) = x$ 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 :$R\left[t\right] \to S, \quad f \mapsto \overline$ (''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 $R\left[t\right]$, there exists a ring ''S'' containing ''R'' such that ''f'' is a product of linear factors in $S\left[t\right]$. 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 $k\left\left[t_1, \ldots, t_n\right\right]$ 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 $R\left[\!\left[t\right]\!\right]$ 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 $R\left[t\right]$ 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).

## 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 $\operatorname_R\left(U\right)$. As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: $\operatorname_R\left(R^n\right) \simeq \operatorname_n\left(R\right)$. 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 $S = \operatorname_R\left(U\right)$, resulting in the ring isomorphism: :$\operatorname_R\left(\oplus_1^n U\right) \to \operatorname_n\left(S\right), \quad f \mapsto \left(f_\right).$ Any ring homomorphism induces . Schur's lemma says that if ''U'' is a simple right ''R''-module, then $\operatorname_R\left(U\right)$ 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 :$\operatorname_R\left(U\right) \simeq \prod_^r \operatorname_ \left(\operatorname_R\left(U_i\right)\right)$. 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 (mathematics), 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: $R\left[t_1, t_2, \cdots\right] = \varinjlim R\left[t_1, t_2, \cdots, t_m\right].$ * The algebraic closure of finite fields of the same characteristic $\overline_p = \varinjlim \mathbf_.$ * The field of formal Laurent series over a field ''k'': $k\left(\!\left(t\right)\!\right) = \varinjlim t^k\left[\!\left[t\right]\!\right]$ (it is the field of fractions of the formal power series ring $k\left[\!\left[t\right]\!\right]$.) * The function field of an algebraic variety over a field ''k'' is $\varinjlim k\left[U\right]$ where the limit runs over all the coordinate rings $k\left[U\right]$ 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 ring, finitely generated subrings. A projective limit (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 $\left(x_n\right)$ 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 of a ring, 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 $R\left[S^\right]$ together with the ring homomorphism $R \to R\left\left[S^\right\right]$ that "inverts" ''S''; that is, the homomorphism maps elements in ''S'' to unit elements in $R\left\left[S^\right\right]$, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through $R\left\left[S^\right\right]$. The ring $R\left\left[S^\right\right]$ 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 $R\left\left[f^\right\right]$ consists of elements of the form $r/f^n, \, r \in R , \, n \ge 0$ (to be precise, $R\left\left[f^\right\right] = R\left[t\right]/\left(tf - 1\right).$) 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 $R\left\left[S^\right\right]$. $R_\mathfrak$ is then a local ring 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 $k\left(\mathfrak\right)$. If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a change of rings $M\left\left[S^\right\right] = R\left\left[S^\right\right] \otimes_R M$. The most important properties of localization are the following: when ''R'' is a commutative ring and ''S'' a multiplicatively closed subset * $\mathfrak \mapsto \mathfrak\left\left[S^\right\right]$ is a bijection between the set of all prime ideals in ''R'' disjoint from ''S'' and the set of all prime ideals in $R\left\left[S^\right\right]$. * $R\left\left[S^\right\right] = \varinjlim R\left\left[f^\right\right]$, ''f'' running over elements in ''S'' with partial ordering given by divisibility. * The localization is exact: *: $0 \to M\text{'}\left\left[S^\right\right] \to M\left\left[S^\right\right] \to M\text{'}\text{'}\left\left[S^\right\right] \to 0$ is exact over $R\left\left[S^\right\right]$ whenever $0 \to M\text{'} \to M \to M\text{'}\text{'} \to 0$ is exact over ''R''. * Conversely, if $0 \to M\text{'}_\mathfrak \to M_\mathfrak \to M\text{'}\text{'}_\mathfrak \to 0$ is exact for any maximal ideal $\mathfrak$, then $0 \to M\text{'} \to M \to M\text{'}\text{'} \to 0$ 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, a localization of a category 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 $R\left\left[S^\right\right]$ and ''R''-modules map to $R\left\left[S^\right\right]$-modules.)

## Completion

Let ''R'' be a commutative ring, and let ''I'' be an ideal of ''R''. The Completion (ring theory), 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 integer, ''p''-adic integers and is denoted Z''p''. The completion can in this case be constructed also from the p-adic absolute value, ''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 $v_p\left(n\right)$ 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 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 $R\left[\right]$ is the completion of $R\left[t\right]$ at $\left(t\right)$ (see also Hensel's lemma) A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, 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 closureIn commutative algebra, an element ''b'' of a commutative ring In ring theory, a branch of abstract algebra, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is ca ...
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 $\left(x \otimes u\right) \left(y \otimes v\right) = xy \otimes uv$. See also: ''Tensor product of algebras'', ''Change of rings''.

# Special kinds of rings

## Domains

A zero ring, nonzero ring with no nonzero zero-divisors is called a domain (ring theory), domain. 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 Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
is on the extent to which the ring of integers, ring of (generalized) integers in a
number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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. 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 $k\left[t\right]$ is a unique factorization domain, ''q'' factors into powers of distinct irreducible polynomials (that is, prime elements): :$q = p_1^ \ldots p_s^.$ Letting $t \cdot v = f\left(v\right)$, 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 $k\left[t\right]/\left\left(p_i^\right\right)$. Now, if $p_i\left(t\right) = t - \lambda_i$, 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 subclass (set theory), 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 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 grassl ...
. 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. A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.

## Semisimple rings

A ''semisimple module'' 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 ring, simple). * For any division ring and positive integer , the matrix ring is semisimple (and simple ring, simple). * For a field and finite group , the group ring is semisimple if and only if the characteristic (algebra), characteristic of does not divide the order (algebra), order of (Maschke's theorem). * Clifford algebras are semisimple. The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a differential operator#Ring of multivariate polynomial differential operators, 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 ring, artinian and semiprimitive ring, semiprimitive. * is a finite direct product $\textstyle \prod_^r \operatorname_\left(D_i\right)$ 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 algebra, 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. 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 $\left[A\right]$ with the multiplication $\left[A\right]\left[B\right] = \left\left[A \otimes_k B\right\right]$ form an abelian group called the Brauer group of ''k'' and is denoted by $\operatorname\left(k\right)$. 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, $\operatorname\left(k\right)$ is trivial if ''k'' is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). $\operatorname\left(\mathbf\right)$ has order 2 (a special case of the Frobenius theorem (real division algebras), theorem of Frobenius). Finally, if ''k'' is a nonarchimedean local field (for example, $\mathbf_p$), then $\operatorname\left(k\right) = \mathbf/\mathbf$ through the Hasse invariant of an algebra, invariant map. Now, if ''F'' is a field extension of ''k'', then the base extension $- \otimes_k F$ induces $\operatorname\left(k\right) \to \operatorname\left(F\right)$. Its kernel is denoted by $\operatorname\left(F/k\right)$. It consists of $\left[A\right]$ 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 $\operatorname\left(F/k\right)$ is canonically isomorphic to $H^2\left\left(\operatorname\left(F/k\right), k^*\right\right)$. Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.

## Valuation ring

If ''K'' is a field, a valuation (algebra), valuation ''v'' is a group homomorphism from the multiplicative group ''K'' to a totally ordered abelian group ''G'' such that, for any ''f'', ''g'' in ''K'' with ''f'' + ''g'' nonzero, The valuation ring of ''v'' is the subring of ''K'' consisting of zero and all nonzero ''f'' such that . Examples: See also: Novikov ring and uniserial ring.

# Rings with extra structure

A ring may be viewed as an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(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 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 if its set of elements ''R'' is given a topological space, topology which makes the addition map ( $+ : R\times R \to R\,$) and the multiplication map ( $\cdot : R\times R \to R\,$) to be both Continuous function (topology), continuous as maps between topological spaces (where ''X'' × ''X'' inherits the product topology or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the Euclidean topology, 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 powers: ::$\lambda^n\left(x + y\right) = \sum_0^n \lambda^i\left(x\right) \lambda^\left(y\right)$. :For example, Z is a λ-ring with $\lambda^n\left(x\right) = \binom$, the binomial coefficients. 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 functor, associated ring.

## Cohomology ring of a topological space

To any topological space ''X'' one can associate its integral
cohomology ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring (mathematics), ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomol ...
:$H^*\left(X,\mathbf\right) = \bigoplus_^ H^i\left(X,\mathbf\right),$ a graded ring. There are also homology groups $H_i\left(X,\mathbf\right)$ of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and torus, tori, for which the methods of point-set topology 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. 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 cup product, 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 variety, algebraic varieties, Schubert calculus and much more.

## Burnside ring of a group

To any
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
is associated its Burnside ring which uses a ring to describe the various ways the group can Group action (mathematics), 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 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
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 is associated its function field of an algebraic variety, function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the
coordinate ring In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
. The study of
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

makes heavy use of commutative algebra 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 polytopes.

# Category-theoretic description

Every ring can be thought of as a monoid (category theory), monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of abelian groups, tensor product of -modules). The monoid action of a ring ''R'' on an abelian group is simply an module (mathematics), ''R''-module. Essentially, an ''R''-module is a generalization of the notion of a vector space – 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 ringIn abstract algebra, the endomorphisms of an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
(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, functor, 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 Group with operators, 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 Product (category theory), 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 $h_R = \operatorname\left(-,R\right) : C^ \to \mathbf$ through the category of rings: $C^ \to \mathbf \stackrel\longrightarrow \mathbf$.

## Ring scheme

In algebraic geometry, a ring scheme over a base Scheme (mathematics), scheme 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, a ring spectrum is a spectrum (topology), spectrum ''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 spectrum, symmetric spectra.

* 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 ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
* Dedekind ring * Differential ring * Exponential field, Exponential ring * Finite ring * Lie ring * Local ring * Noetherian ring, Noetherian and artinian rings * Ordered ring * Poisson ring * Reduced ring * Regular ring * Ring of periods * SBI ring * Valuation ring and discrete valuation ring

# References

## General references

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

## Special references

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

* * *

## 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) Bronshtein and Semendyayev, 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 . * Itô, K. editor (1986) "Rings." §368 in ''Encyclopedic Dictionary of Mathematics'', 2nd ed., Vol. 2. Cambridge, MA: MIT Press. * Israel Kleiner (mathematician), 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 (1985) ''A History of Algebra'', Springer-Verlag, {{DEFAULTSORT:Ring (Mathematics) Algebraic structures Ring theory