In

/ref> In 1871,

^{''n''} is going to be an integral linear combination of 1, ''a'', and ''a''^{2}.

_{''R''}(''V'').
* If ''G'' is 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

_{''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 $R\; \backslash simeq\; S$. 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 $R\; \backslash to\; R,\; x\; \backslash mapsto\; uxu^$ is a ring homomorphism, called an inner automorphism of ''R''.
* Let ''R'' be a commutative ring of prime characteristic ''p''. Then $x\; \backslash mapsto\; x^p$ 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 $f:R\; \backslash to\; S$, 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).

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.
* $a(x+y)=ax+ay$
* $(a+b)x=ax+bx$
* $1x=x$
* $(ab)x=a(bx)$
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

_{1}, ''s''_{1}) + (''r''_{2}, ''s''_{2}) = (''r''_{1} + ''r''_{2}, ''s''_{1} + ''s''_{2})
* (''r''_{1}, ''s''_{1}) ⋅ (''r''_{2}, ''s''_{2}) = (''r''_{1} ⋅ ''r''_{2}, ''s''_{1} ⋅ ''s''_{2})
for all ''r''_{1}, ''r''_{2} in ''R'' and ''s''_{1}, ''s''_{2} in ''S''. The ring with the above operations of addition and multiplication and the multiplicative identity $(1,\; 1)$ is called the Direct product 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 $\backslash textstyle\; \backslash prod\_\; R\_i$ is a ring with componentwise addition and multiplication.
Let ''R'' be a commutative ring and $\backslash mathfrak\_1,\; \backslash cdots,\; \backslash mathfrak\_n$ be ideals such that $\backslash mathfrak\_i\; +\; \backslash mathfrak\_j\; =\; (1)$ whenever $i\; \backslash ne\; j$. Then the Chinese remainder theorem says there is a canonical ring isomorphism:
:$R\; /\; \backslash simeq\; \backslash prod\_^,\; \backslash qquad\; x\; \backslash ;\backslash operatorname\backslash ;\; \backslash mapsto\; (x\; \backslash ;\backslash operatorname\backslash ;\; \backslash mathfrak\_1,\; \backslash ldots\; ,\; x\; \backslash ;\backslash operatorname\backslash ;\; \backslash mathfrak\_n)$.
A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let $R\_i,\; 1\; \backslash le\; i\; \backslash le\; n$ be rings, $\backslash textstyle\; R\_i\; \backslash to\; R\; =\; \backslash prod\; R\_i$ the inclusions with the images $\backslash mathfrak\_i$ (in particular $\backslash mathfrak\_i$ are rings though not subrings). Then $\backslash mathfrak\_i$ are ideals of ''R'' and
:$R\; =\; \backslash mathfrak\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash mathfrak\_n,\; \backslash quad\; \backslash mathfrak\_i\; \backslash mathfrak\_j\; =\; 0,\; i\; \backslash ne\; j,\; \backslash quad\; \backslash mathfrak\_i^2\; \backslash subseteq\; \backslash mathfrak\_i$
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
:$1\; =\; e\_1\; +\; \backslash cdots\; +\; e\_n,\; \backslash quad\; e\_i\; \backslash in\; \backslash mathfrak\_i.$
By the conditions on $\backslash mathfrak\_i$, one has that $e\_i$ are central idempotents and $e\_i\; e\_j\; =\; 0,\; i\; \backslash ne\; j$ (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let $\backslash mathfrak\_i\; =\; R\; e\_i$, 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).

^{2} and ''t''^{3}.
Example: let ''f'' be a polynomial in one variable, that is, an element in a polynomial ring ''R''. Then $f(x+h)$ is an element in $R[h]$ and $f(x\; +\; h)\; -\; f(x)$ is divisible by ''h'' in that ring. The result of substituting zero to ''h'' in $(f(x\; +\; h)\; -\; f(x))/h$ is $f\text{'}(x)$, the derivative of ''f'' at ''x''.
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism $\backslash phi:\; R\; \backslash to\; S$ and an element ''x'' in ''S'' there exists a unique ring homomorphism $\backslash overline:\; R[t]\; \backslash to\; S$ such that $\backslash overline(t)\; =\; x$ and $\backslash overline$ restricts to $\backslash phi$. 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 $R\; \backslash to\; S$. The universal property says that this map extends uniquely to
:$R[t]\; \backslash to\; S,\; \backslash quad\; f\; \backslash mapsto\; \backslash overline$
(''t'' maps to ''x'') where $\backslash overline$ 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 $R[t]$, there exists a ring ''S'' containing ''R'' such that ''f'' is a product of linear factors in $S[t]$.
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 $k\backslash left[t\_1,\; \backslash ldots,\; t\_n\backslash right]$ 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 $R[\backslash ![t]\backslash !]$ consists of formal power series
: $\backslash sum\_0^\backslash infty\; a\_i\; t^i,\; \backslash quad\; a\_i\; \backslash in\; R$
together with multiplication and addition that mimic those for convergent series. It contains $R[t]$ 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).

_{''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 $\backslash operatorname\_R(U)$.
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: $\backslash operatorname\_R(R^n)\; \backslash simeq\; \backslash operatorname\_n(R)$. This is a special case of the following fact: If $f:\; \backslash oplus\_1^n\; U\; \backslash to\; \backslash oplus\_1^n\; U$ is an ''R''-linear map, then ''f'' may be written as a matrix with entries $f\_$ in $S\; =\; \backslash operatorname\_R(U)$, resulting in the ring isomorphism:
:$\backslash operatorname\_R(\backslash oplus\_1^n\; U)\; \backslash to\; \backslash operatorname\_n(S),\; \backslash quad\; f\; \backslash mapsto\; (f\_).$
Any ring homomorphism induces .
Schur's lemma says that if ''U'' is a simple right ''R''-module, then $\backslash operatorname\_R(U)$ is a division ring. If $\backslash textstyle\; U\; =\; \backslash bigoplus\_^r\; U\_i^$ is a direct sum of ''m''_{''i''}-copies of simple ''R''-modules $U\_i$, then
:$\backslash operatorname\_R(U)\; \backslash simeq\; \backslash prod\_^r\; \backslash operatorname\_\; (\backslash operatorname\_R(U\_i))$.
The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.
A ring ''R'' and the matrix ring M_{''n''}(''R'') over it are Morita equivalent: the Category (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'').

_{''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 $\backslash varinjlim\; R\_i$ defined as follows: it is the disjoint union of all ''R''_{''i''}'s modulo the equivalence relation $x\; \backslash sim\; y$ if and only if $x\; =\; y$ in ''R''_{''i''} for sufficiently large ''i''.
Examples of colimits:
* A polynomial ring in infinitely many variables: $R[t\_1,\; t\_2,\; \backslash cdots]\; =\; \backslash varinjlim\; R[t\_1,\; t\_2,\; \backslash cdots,\; t\_m].$
* The algebraic closure of finite fields of the same characteristic $\backslash overline\_p\; =\; \backslash varinjlim\; \backslash mathbf\_.$
* The field of formal Laurent series over a field ''k'': $k(\backslash !(t)\backslash !)\; =\; \backslash varinjlim\; t^k[\backslash ![t]\backslash !]$ (it is the field of fractions of the formal power series ring $k[\backslash ![t]\backslash !]$.)
* The function field of an algebraic variety over a field ''k'' is $\backslash varinjlim\; k[U]$ where the limit runs over all the coordinate rings $k[U]$ of nonempty open subsets ''U'' (more succinctly it is the stalk (mathematics), stalk of the structure sheaf at the generic point.)
Any commutative ring is the colimit of finitely generated 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 $R\_j\; \backslash to\; R\_i,\; j\; \backslash ge\; i$ such that $R\_i\; \backslash to\; R\_i$ are all the identities and $R\_k\; \backslash to\; R\_j\; \backslash to\; R\_i$ is $R\_k\; \backslash to\; R\_i$ whenever $k\; \backslash ge\; j\; \backslash ge\; i$. Then $\backslash varprojlim\; R\_i$ is the subring of $\backslash textstyle\; \backslash prod\; R\_i$ consisting of $(x\_n)$ such that $x\_j$ maps to $x\_i$ under $R\_j\; \backslash to\; R\_i,\; j\; \backslash ge\; i$.
For an example of a projective limit, see .

_{''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 $x\; \backslash mapsto\; ,\; x,$ from Q to R given by $,\; n,\; \_p=p^$ where $v\_p(n)$ denotes the exponent of ''p'' in the prime factorization of a nonzero integer ''n'' into prime numbers (we also put $,\; 0,\; \_p=0$ and $,\; m/n,\; \_p\; =\; ,\; m,\; \_p/,\; n,\; \_p$). It defines a distance function on Q and the completion of Q as a metric space 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 $,\; x,\; \_p\; \backslash le\; 1$ is isomorphic to Z_{''p''}.
Similarly, the formal power series ring $R[]$ is the completion of $R[t]$ at $(t)$ (see also Hensel's lemma)
A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, 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

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.

^{∗} 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.

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 ( $+\; :\; R\backslash times\; R\; \backslash to\; R\backslash ,$) and the multiplication map ( $\backslash cdot\; :\; R\backslash times\; R\; \backslash to\; R\backslash ,$) 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:
::$\backslash lambda^n(x\; +\; y)\; =\; \backslash sum\_0^n\; \backslash lambda^i(x)\; \backslash lambda^(y)$.
:For example, Z is a λ-ring with $\backslash lambda^n(x)\; =\; \backslash binom$, 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.

_{''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.

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

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
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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 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 ...

.
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 anabelian 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 iscommutative
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 nonzero
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 $\backslash mathbf$, consisting of thenumber
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 anabsorbing 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 $\backslash mathbf/4\backslash mathbf\; =\; \backslash left\backslash $ with the following operations: * The sum $\backslash overline\; +\; \backslash overline$ 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, $\backslash overline\; +\; \backslash overline\; =\; \backslash overline$ and $\backslash overline\; +\; \backslash overline\; =\; \backslash overline$. * The product $\backslash overline\; \backslash cdot\; \backslash overline$ in Z/4Z is the remainder when the integer ''xy'' is divided by 4. For example, $\backslash overline\; \backslash cdot\; \backslash overline\; =\; \backslash overline$ and $\backslash overline\; \backslash cdot\; \backslash overline\; =\; \backslash overline$. 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 $\backslash overline$, which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any $\backslash overline$ in Z/4Z is $\backslash overline$. For example, $-\backslash overline\; =\; \backslash overline\; =\; \backslash overline.$Example: 2-by-2 matrices

The set of 2-by-2square 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
:$\backslash operatorname\_2(F)\; =\; \backslash left\backslash .$
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 ...

, $\backslash operatorname\_2(F)$ satisfies the above ring axioms. The element $\backslash left(\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\backslash right)$ is the multiplicative identity of the ring. If $A\; =\; \backslash left(\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 1\; \&\; 0\; \backslash end\; \backslash right)$ and $B\; =\; \backslash left(\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\; \backslash right)$, then $AB\; =\; \backslash left(\; \backslash begin\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; \backslash right)$ while $BA\; =\; \backslash left(\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \backslash end\; \backslash right)$; 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 ofpolynomial 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 byDavid 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''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 amultiplicative 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 calledfields
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 ...

.
* 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 .
* $\backslash mathbf;\; href="/html/ALL/s/.html"\; ;"title="">$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.
* $\backslash mathbf\backslash left;\; href="/html/ALL/s/left(1\_+\_\backslash sqrt\backslash right)/2\backslash right.html"\; ;"title="left(1\; +\; \backslash sqrt\backslash right)/2\backslash right">left(1\; +\; \backslash sqrt\backslash right)/2\backslash right$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 ...

''.
* $\backslash mathbf;\; href="/html/ALL/s/.html"\; ;"title="">$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.
* $\backslash mathbf;\; href="/html/ALL/s/left(1\_+\_\backslash sqrt\backslash right)/2.html"\; ;"title="left(1\; +\; \backslash sqrt\backslash right)/2">left(1\; +\; \backslash sqrt\backslash right)/2$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 Endgroup
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.
at genezanazwisk.pl (the webpage cites the sources)

Non-rings

Basic concepts

Products and powers

For each nonnegative integer , given a sequence $(a\_1,\backslash ldots,a\_n)$ of elements of , one can define the product $\backslash textstyle\; P\_n\; =\; \backslash prod\_^n\; a\_i$ 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 leftzero 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 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 $R^\backslash times$ or $R^*$ or $U(R)$. For example, if ''R'' is the ring of all square matrices of size ''n'' over a field, then $R^\backslash times$ 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 asubring
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 $n\backslash cdot\; 1=1+1+\backslash ldots+1$ (''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, $n\backslash cdot\; 1$ is never zero for any positive integer ''n'', and those rings are said to have ''characteristic zero''.
Given a ring ''R'', let $\backslash operatorname(R)$ denote the set of all elements ''x'' in ''R'' such that ''x'' commutes with every element in ''R'': $xy\; =\; yx$ for any ''y'' in ''R''. Then $\backslash operatorname(R)$ 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 :$r\_1\; x\_1\; +\; \backslash cdots\; +\; r\_n\; x\_n\; \backslash quad\; \backslash textrm\backslash ;\backslash textrm\backslash ;\; r\_i\; \backslash in\; R\; \backslash ;\; \backslash textrm\; \backslash ;\; x\_i\; \backslash in\; I,$ then ''I'' is a left ideal if $R\; I\; \backslash subseteq\; I$. Similarly, a right ideal is a subset ''I'' such that $I\; R\; \backslash subseteq\; I$. 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 $(x)$. For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be simple 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 $x,\; y\backslash in\; R$ we have that $xy\; \backslash in\; P$ implies either $x\; \backslash in\; P$ or $y\backslash in\; P$. Equivalently, ''P'' is prime if for any ideals $I,\; J$ we have that $IJ\; \backslash subseteq\; P$ implies either $I\; \backslash subseteq\; P$ or $J\; \backslash subseteq\; P.$ 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''(1Quotient 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 $p\; \backslash colon\; R\; \backslash to\; R/I$, given by $x\; \backslash mapsto\; x\; +\; I$. It is surjective and satisfies the following universal property: *If $f\; \backslash colon\; R\; \backslash to\; S$ is a ring homomorphism such that $f(I)\; =\; 0$, then there is a unique homomorphism $\backslash overline\; \backslash colon\; R/I\; \backslash to\; S$ such that $f\; =\; \backslash overline\; \backslash circ\; p$. For any ring homomorphism $f\; \backslash colon\; R\; \backslash to\; S$, invoking the universal property with $I\; =\; \backslash ker\; f$ produces a homomorphism $\backslash overline\; \backslash colon\; R/\backslash ker\; f\; \backslash to\; S$ that gives an isomorphism from $R/\backslash ker\; f$ to the image of .Module

The concept of a ''module over a ring'' generalizes the concept of a vector space (over aadditive 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''Polynomial ring

Given a symbol ''t'' (called a variable) and a commutative ring ''R'', the set of polynomials : $R[t]\; =\; \backslash left\backslash $ forms a commutative ring with the usual addition and multiplication, containing ''R'' as a subring. It is called thepolynomial 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 $R\backslash left[t\_1,\; \backslash ldots,\; t\_n\backslash right]$ of all polynomials in variables $t\_1,\; \backslash ldots,\; t\_n$ forms a commutative ring, containing $R\backslash left[t\_i\backslash right]$ as subrings.
If ''R'' is an integral domain, then $R[t]$ is also an integral domain; its field of fractions is the field of rational functions. If ''R'' is a Noetherian ring, then $R[t]$ is a Noetherian ring. If ''R'' is a unique factorization domain, then $R[t]$ is a unique factorization domain. Finally, ''R'' is a field if and only if $R[t]$ is a principal ideal domain.
Let $R\; \backslash subseteq\; S$ be commutative rings. Given an element ''x'' of ''S'', one can consider the ring homomorphism
: $R[t]\; \backslash to\; S,\; \backslash quad\; f\; \backslash mapsto\; f(x)$
(that is, the substitution (algebra), substitution). If and , then . Because of this, the polynomial ''f'' is often also denoted by $f(t)$. The image of the map $f\; \backslash mapsto\; f(x)$ is denoted by $R[x]$; it is the same thing as the subring of ''S'' generated by ''R'' and ''x''.
Example: $k\backslash left[t^2,\; t^3\backslash right]$ denotes the image of the homomorphism
:$k[x,\; y]\; \backslash to\; k[t],\; \backslash ,\; f\; \backslash mapsto\; f\backslash left(t^2,\; t^3\backslash right).$
In other words, it is the subalgebra of $k[t]$ generated by ''t''Matrix ring and endomorphism ring

Let ''R'' be a ring (not necessarily commutative). The set of all square matrices of size ''n'' with entries in ''R'' forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by MLimits and colimits of rings

Let ''R''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 $R[S^]$ together with the ring homomorphism $R\; \backslash to\; R\backslash left[S^\backslash right]$ that "inverts" ''S''; that is, the homomorphism maps elements in ''S'' to unit elements in $R\backslash left[S^\backslash right]$, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through $R\backslash left[S^\backslash right]$. The ring $R\backslash left[S^\backslash right]$ 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 $R\backslash left[f^\backslash right]$ consists of elements of the form $r/f^n,\; \backslash ,\; r\; \backslash in\; R\; ,\; \backslash ,\; n\; \backslash ge\; 0$ (to be precise, $R\backslash left[f^\backslash right]\; =\; R[t]/(tf\; -\; 1).$) The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in ''R''. In that case $S\; =\; R\; -\; \backslash mathfrak$, one often writes $R\_\backslash mathfrak$ for $R\backslash left[S^\backslash right]$. $R\_\backslash mathfrak$ is then a local ring with the maximal ideal $\backslash mathfrak\; R\_\backslash mathfrak$. 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 $\backslash mathfrak$ is a prime ideal of a commutative ring ''R'', then the field of fractions of $R/\backslash mathfrak$ is the same as the residue field of the local ring $R\_\backslash mathfrak$ and is denoted by $k(\backslash mathfrak)$. If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a change of rings $M\backslash left[S^\backslash right]\; =\; R\backslash left[S^\backslash right]\; \backslash otimes\_R\; M$. The most important properties of localization are the following: when ''R'' is a commutative ring and ''S'' a multiplicatively closed subset * $\backslash mathfrak\; \backslash mapsto\; \backslash mathfrak\backslash left[S^\backslash right]$ is a bijection between the set of all prime ideals in ''R'' disjoint from ''S'' and the set of all prime ideals in $R\backslash left[S^\backslash right]$. * $R\backslash left[S^\backslash right]\; =\; \backslash varinjlim\; R\backslash left[f^\backslash right]$, ''f'' running over elements in ''S'' with partial ordering given by divisibility. * The localization is exact: *: $0\; \backslash to\; M\text{'}\backslash left[S^\backslash right]\; \backslash to\; M\backslash left[S^\backslash right]\; \backslash to\; M\text{'}\text{'}\backslash left[S^\backslash right]\; \backslash to\; 0$ is exact over $R\backslash left[S^\backslash right]$ whenever $0\; \backslash to\; M\text{'}\; \backslash to\; M\; \backslash to\; M\text{'}\text{'}\; \backslash to\; 0$ is exact over ''R''. * Conversely, if $0\; \backslash to\; M\text{'}\_\backslash mathfrak\; \backslash to\; M\_\backslash mathfrak\; \backslash to\; M\text{'}\text{'}\_\backslash mathfrak\; \backslash to\; 0$ is exact for any maximal ideal $\backslash mathfrak$, then $0\; \backslash to\; M\text{'}\; \backslash to\; M\; \backslash to\; M\text{'}\text{'}\; \backslash to\; 0$ 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 $R\backslash left[S^\backslash right]$ and ''R''-modules map to $R\backslash left[S^\backslash right]$-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 $\backslash hat\; =\; \backslash varprojlim\; R/I^n$; it is a commutative ring. The canonical homomorphisms from ''R'' to the quotients $R/I^n$ induce a homomorphism $R\; \backslash to\; \backslash hat$. 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 Zintegral 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 $F\; \backslash to\; R$ 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\; =\; \backslash $, 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 $S\; \backslash mapsto\; \backslash text\; S$ 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 $A\; \backslash otimes\_R\; B$ is an ''R''-algebra with multiplication characterized by $(x\; \backslash otimes\; u)\; (y\; \backslash otimes\; v)\; =\; xy\; \backslash otimes\; uv$. 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 inalgebraic 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 $f:\; V\; \backslash to\; V$ a linear map with minimal polynomial ''q''. Then, since $k[t]$ is a unique factorization domain, ''q'' factors into powers of distinct irreducible polynomials (that is, prime elements):
:$q\; =\; p\_1^\; \backslash ldots\; p\_s^.$
Letting $t\; \backslash cdot\; v\; =\; f(v)$, we make ''V'' a ''k''[''t'']-module. The structure theorem then says ''V'' is a direct sum of cyclic modules, each of which is isomorphic to the module of the form $k[t]/\backslash left(p\_i^\backslash right)$. Now, if $p\_i(t)\; =\; t\; -\; \backslash lambda\_i$, 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 $t\; -\; \backslash lambda\_i$ 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 aSemisimple 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 $\backslash textstyle\; \backslash prod\_^r\; \backslash operatorname\_(D\_i)$ where each is a positive integer, and each is a division ring (Artin–Wedderburn theorem). Semisimplicity is closely related to separability. A unital associative algebra ''A'' over a field ''k'' is said to be separable algebra, separable if the base extension $A\; \backslash otimes\_k\; F$ 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 $A\; \backslash otimes\_k\; k\_n\; \backslash approx\; B\; \backslash otimes\_k\; k\_m$. Since $k\_n\; \backslash otimes\_k\; k\_m\; \backslash simeq\; k\_$, the similarity is an equivalence relation. The similarity classes $[A]$ with the multiplication $[A][B]\; =\; \backslash left[A\; \backslash otimes\_k\; B\backslash right]$ form an abelian group called the Brauer group of ''k'' and is denoted by $\backslash operatorname(k)$. By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, $\backslash operatorname(k)$ is trivial if ''k'' is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). $\backslash operatorname(\backslash mathbf)$ 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, $\backslash mathbf\_p$), then $\backslash operatorname(k)\; =\; \backslash mathbf/\backslash mathbf$ through the Hasse invariant of an algebra, invariant map. Now, if ''F'' is a field extension of ''k'', then the base extension $-\; \backslash otimes\_k\; F$ induces $\backslash operatorname(k)\; \backslash to\; \backslash operatorname(F)$. Its kernel is denoted by $\backslash operatorname(F/k)$. It consists of $[A]$ such that $A\; \backslash otimes\_k\; F$ is a matrix ring over ''F'' (that is, ''A'' is split by ''F''.) If the extension is finite and Galois, then $\backslash operatorname(F/k)$ is canonically isomorphic to $H^2\backslash left(\backslash operatorname(F/k),\; k^*\backslash right)$. 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''Rings with extra structure

A ring may be viewed as anSome 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 integralcohomology 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 ...

:$H^*(X,\backslash mathbf)\; =\; \backslash bigoplus\_^\; H^i(X,\backslash mathbf),$
a graded ring. There are also homology groups $H\_i(X,\backslash mathbf)$ 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 anygroup
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 anygroup 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 thecoordinate 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 itsendomorphism 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 EndGeneralization

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 $\backslash $ 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 $R\; \backslash times\; R\backslash ;\backslash stackrel\backslash to\backslash ,R$ (addition), $R\; \backslash times\; R\backslash ;\backslash stackrel\backslash to\backslash ,R$ (multiplication), $\backslash operatorname\backslash stackrel\backslash to\backslash ,R$ (additive identity), $R\backslash ;\backslash stackrel\backslash to\backslash ,R$ (additive inverse), and $\backslash operatorname\backslash stackrel\backslash to\backslash ,R$ (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object ''R'' equipped with a factorization of its functor of points $h\_R\; =\; \backslash operatorname(-,R)\; :\; C^\; \backslash to\; \backslash mathbf$ through the category of rings: $C^\; \backslash to\; \backslash mathbf\; \backslash stackrel\backslash longrightarrow\; \backslash mathbf$.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 $\backslash mu\; \backslash colon\; X\; \backslash wedge\; X\; \backslash to\; X$ and a unit map $S\; \backslash to\; X$ 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.See also

* Algebra over a commutative ring * Categorical ring * Category of rings * Glossary of ring theory * Nonassociative ring * Ring of sets * Semiring * Spectrum of a ring * Simplicial commutative ring Special types of rings: *Boolean 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
Notes

Citations

References

General references

* * * * * . * * * * * * * * * . * * * * . * * * . * . * * *Special references

* * * * * * * * * * * * * * * * * * * *Primary sources

* * *Historical references

History of ring theory at the MacTutor Archive

* Garrett Birkhoff and Saunders Mac Lane (1996) ''A Survey of Modern Algebra'', 5th ed. New York: Macmillan. * Bronshtein, I. N. and Semendyayev, K. A. (2004) 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