Ring (mathematics)


In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.
A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication.
Whether a ring is commutative or not has profound implications on its behavior as an abstract object. As a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has been greatly influenced by problems and ideas occurring naturally in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.
The conceptualization of rings began in the 1870s and was completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they also proved to be useful in other branches of mathematics such as geometry and mathematical analysis.

Definition and illustration

The most familiar example of a ring is the set of all integers,, consisting of the numbers
The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings.

Definition

A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms
  1. R is an abelian group under addition, meaning that:
  2. * + c = a + for all a, b, c in R .
  3. * a + b = b + a for all a, b in R .
  4. * There is an element 0 in R such that a + 0 = a for all a in R .
  5. * For each a in R there exists −a in R such that a + = 0 .
  6. R is a monoid under multiplication, meaning that:
  7. * · c = a · for all a, b, c in R .
  8. * There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R .
  9. Multiplication is distributive with respect to addition, meaning that:
  10. * a ⋅ = + for all a, b, c in R .
  11. * · a = + for all a, b, c in R .

    Basic properties

Some basic properties of a ring follow immediately from the axioms:
Equip the set with the following operations:
Then Z4 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 Z4, and this element is often denoted by or, which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any in Z4 is. For example,

Example: 2-by-2 matrices

The set of 2-by-2 matrices with real number entries is written
With the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms. The element is the multiplicative identity of the ring. If and, then while ; this example shows that the ring is noncommutative.
More generally, for any ring R, commutative or not, and any nonnegative integer n, one may form the ring of n-by-n matrices with entries in R: see Matrix ring.

History

Dedekind

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" and "module" and studied their properties. But Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

Hilbert

The term "Zahlring" was coined by David Hilbert in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense, 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. 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, an is going to be an integral linear combination of 1, a, and a2.

Fraenkel and Noether

The first axiomatic definition of a ring was given by Adolf Fraenkel in 1914, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave the modern axiomatic definition of ring and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.

Multiplicative identity: mandatory vs. optional

Fraenkel required a ring to have a multiplicative identity 1, whereas Noether did not.
Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1. 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. But even today, there are many books that do not require a 1.
Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms.
In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that rings without a multiplicative identity are not totally associative 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".
In the second category, we find authors who use the following terms:

Basic examples

Commutative rings

Basic concepts

Elements in a ring

A left zero divisor of a ring is a nonzero element in the ring such that there exists a nonzero element of such that. A right zero divisor is defined similarly.
A nilpotent element is an element such that for some. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.
An idempotent is an element such that. One example of an idempotent element is a projection in linear algebra.
A unit is an element having a multiplicative inverse; in this case the inverse is unique, and is denoted by. The set of units of a ring is a group under ring multiplication; this group is denoted by or or. For example, if R is the ring of all square matrices of size n over a field, then consists of the set of all invertible matrices of size n, and is called the general linear group.

Subring

A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S. Equivalently, S is a subring if it is not empty, and for any x, y in S,, and are in S. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R. So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring.
For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z. 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.
An intersection of subrings is a subring. The smallest subring containing a given subset E of R is called a subring generated by E. Such a subring exists since it is the intersection of all subrings containing E.
For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and −1 together many times in any mixture. It is possible that can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R. In some rings, is never zero for any positive integer n, and those rings are said to have characteristic zero.
Given a ring R, let denote the set of all elements x in R such that x commutes with every element in R: for any y in R. Then is a subring of R; called the center of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the centralizer 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 generate a subring of the center.

Ideal

The definition of an ideal in a ring is analogous to that of normal subgroup in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, the name "ideal". Like elements of rings, the study of ideals is central to structural understanding of a ring.
Let R be a ring. A nonempty subset I of R is then said to be a left ideal in R if, for any x, y in I and r in R, and are in I. If denotes the span of I over R, that is, the set of finite sums
then I is a left ideal if. Similarly, I is said to be right ideal if. 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 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 and are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal is written as. 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 if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.
Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian. The 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 we have that implies either or. Equivalently, P is prime if for any ideals we have that implies either or This latter formulation illustrates the idea of ideals as generalizations of elements.

Homomorphism

A 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:
If one is working with not necessarily unital rings, then the third condition is dropped.
A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to f. Any bijective ring homomorphism is a ring isomorphism. Two rings are said to be isomorphic if there is an isomorphism between them and in that case one writes. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism.
Examples:
Given a ring homomorphism, the set of all elements mapped to 0 by f is called the kernel of f. The kernel is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.
To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A.

Quotient ring

The quotient ring of a ring is analogous to the notion of a quotient group of a group. More formally, given a ring and a two-sided ideal I of, the quotient ring R/I is the set of cosets of I, that is, cosets with respect to ) together with the operations:
for all a, b in R.
Like the case of a quotient group, there is a canonical map given by. It is surjective and satisfies the universal property: if is a ring homomorphism such that, then there is a unique
such that. In particular, taking I to be the kernel, one sees that the quotient ring is isomorphic to the image of f; the fact known as the first isomorphism theorem. The last fact implies that actually any surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.

Module

The concept of a module over a ring generalizes the concept of a vector space by generalizing from multiplication of vectors with elements of a field to multiplication with elements of a ring. More precisely, given a ring with 1, an -module is an abelian group equipped with an operation that satisfies certain axioms. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all in and all in, we have:
When the ring is 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 becomes, if left multiplication 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 by a single invariant. In particular, not all modules have a basis.
The axioms of modules imply that, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
Any ring homomorphism induces a structure of a module: if is a ring homomorphism, then is a left module over by the multiplication:. If is commutative or if is contained in the center of, the ring is called a -algebra. In particular, every ring is an algebra over the integers.

Constructions

Direct product

Let R and S be rings. Then the product can be equipped with the following natural ring structure:
for all r1, r2 in R and s1, s2 in S. The ring with the above operations of addition and multiplication and the multiplicative identity is called the direct product of R with S. The same construction also works for an arbitrary family of rings: if are rings indexed by a set I, then is a ring with componentwise addition and multiplication.
Let R be a commutative ring and be ideals such that whenever. Then the Chinese remainder theorem says there is a canonical ring isomorphism:
A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let be rings, the inclusions with the images . Then are ideals of R and
as a direct sum of abelian groups. 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 R has the above decomposition. Then we can write
By the conditions on, one has that are central idempotents and . Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let, which are two-sided ideals. If each 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. Another application is a restricted product of a family of rings.

Polynomial ring

Given a symbol t and a commutative ring R, the set of polynomials
forms a commutative ring with the usual addition and multiplication, containing R as a subring. It is called the polynomial ring over R. More generally, the set of all polynomials in variables forms a commutative ring, containing as subrings.
If R is an integral domain, then is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, then is a Noetherian ring. If R is a unique factorization domain, then is a unique factorization domain. Finally, R is a field if and only if is a principal ideal domain.
Let be commutative rings. Given an element x of S, one can consider the ring homomorphism
. If S = R and x = t, then f = f. Because of this, the polynomial f is often also denoted by. The image of the map is denoted by ; it is the same thing as the subring of S generated by R and x.
Example: denotes the image of the homomorphism
In other words, it is the subalgebra of generated by t2 and t3.
Example: let f be a polynomial in one variable, that is, an element in a polynomial ring R. Then is an element in and is divisible by h in that ring. The result of substituting zero to h in is, 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 and an element x in S there exists a unique ring homomorphism such that and restricts to. 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. The universal property says that this map extends uniquely to
where 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, there exists a ring S containing R such that f is a product of linear factors in.
Let k be an algebraically closed field. The Hilbert's Nullstellensatz states that there is a natural one-to-one correspondence between the set of all prime ideals in and the set of closed subvarieties of. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring.
There are some other related constructions. A formal power series ring consists of formal power series
together with multiplication and addition that mimic those for convergent series. It contains 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.

Matrix ring and endomorphism ring

Let R be a ring. 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 Mn. Given a right R-module, 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.
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring:. This is a special case of the following fact: If is an R-linear map, then f may be written as a matrix with entries in, resulting in the ring isomorphism:
Any ring homomorphism R → S induces ; in fact, any ring homomorphism between matrix rings arises in this way.
Schur's lemma says that if U is a simple right R-module, then is a division ring. If is a direct sum of mi-copies of simple R-modules, then
The Artin–Wedderburn theorem states any semisimple ring is of this form.
A ring R and the matrix ring Mn over it are Morita equivalent: the category of right modules of R is equivalent to the category of right modules over Mn. In particular, two-sided ideals in R correspond in one-to-one to two-sided ideals in Mn.
Examples:
Let Ri be a sequence of rings such that Ri is a subring of Ri+1 for all i. Then the union of Ri is the ring defined as follows: it is the disjoint union of all Ri's modulo the equivalence relation if and only if in Ri for sufficiently large i.
Examples of colimits:
Any commutative ring is the colimit of finitely generated subrings.
A projective limit of rings is defined as follows. Suppose we're given a family of rings, i running over positive integers, say, and ring homomorphisms such that are all the identities and is whenever. Then is the subring of consisting of such that maps to under.
For an example of a projective limit, see.

Localization

The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a ring R and a subset S of R, there exists a ring together with the ring homomorphism that "inverts" S; that is, the homomorphism maps elements in S to unit elements in, and, moreover, any ring homomorphism from R that "inverts" S uniquely factors through. The ring 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 consists of elements of the form
The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal in R. In that case, one often writes for. is then a local ring with the maximal ideal. 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 is a prime ideal of a commutative ring R, then the field of fractions of is the same as the residue field of the local ring and is denoted by.
If M is a left R-module, then the localization of M with respect to S is given by a change of rings.
The most important properties of localization are the following: when R is a commutative ring and S a multiplicatively closed subset
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.

Completion

Let R be a commutative ring, and let I be an ideal of R.
The completion of R at I is the projective limit ; it is a commutative ring. The canonical homomorphisms from R to the quotients induce a homomorphism. 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 Zp of Z at the principal ideal generated by a prime number p; it is called the ring of p-adic integers. The completion can in this case be constructed also from the p-adic absolute value on Q. The p-adic absolute value on Q is a map from Q to R given by where denotes the exponent of p in the prime factorization of a nonzero integer n into prime numbers. It defines a distance function on Q and the completion of Q as a metric space is denoted by Qp. It is again a field since the field operations extend to the completion. The subring of Qp consisting of elements x with is isomorphic to Zp.
Similarly, the formal power series ring is the completion of at
A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.

Rings with generators and relations

The most general way to construct a ring is by specifying generators and relations. Let F be a free ring 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 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, then the resulting ring will be the usual polynomial ring with coefficients in A in variables that are elements of X
In the category-theoretic terms, the formation is the left adjoint functor of the forgetful functor from the category of rings to Set
Let A, B be algebras over a commutative ring R. Then the tensor product of R-modules is a R-module. We can turn it to a ring by extending linearly. See also: tensor product of algebras, change of rings.

Special kinds of rings

Domains

A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideals domains, PID 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, an integral domain in which every nonunit element is a product of prime elements The fundamental question in algebraic number theory is on the extent to which the ring of integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.
Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. 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 a linear map with minimal polynomial q. Then, since is a unique factorization domain, q factors into powers of distinct irreducible polynomials :
Letting, we make V a k-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. Now, if, then such a cyclic module has a basis in which the restriction of f is represented by a Jordan matrix. Thus, if, say, k is algebraically closed, then all 's are of the form 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 is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.
The following is a chain of class inclusions that describes the relationship between rings, domains and fields:

Division ring

A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain is a field; in particular commutative.
Every module over a division ring is a free module ; 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. Cartan famously asked the following question: given a division ring D and a proper sub-division-ring S that is not contained in the center, does each inner automorphism of D restrict to an automorphism of S? The answer is negative: this is the Cartan–Brauer–Hua theorem.
A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.

Semisimple rings

A ring is called a semisimple ring if it is semisimple as a left module over itself, that is, a direct sum of simple modules. A ring is called a semiprimitive ring if its Jacobson radical is zero. A ring is semisimple if and only if it is artinian and is semiprimitive.
An algebra over a field k is artinian if and only if it has finite dimension. Thus, a semisimple algebra over a field is necessarily finite-dimensional, while a simple algebra may have infinite dimension, for example, the ring of differential operators.
Any module over a semisimple ring is semisimple.
Examples of semisimple rings:
Semisimplicity is closely related to separability. An algebra A over a field k is said to be separable if the base extension is semisimple for any field extension. If A happens to be a field, then this is equivalent to the usual definition in field theory

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.
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. Since, the similarity is an equivalence relation. The similarity classes with the multiplication form an abelian group called the Brauer group of k and is denoted by. 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, is trivial if k is a finite field or an algebraically closed field. has order 2. Finally, if k is a nonarchimedean local field, then through the invariant map.
Now, if F is a field extension of k, then the base extension induces. Its kernel is denoted by. It consists of such that is a matrix ring over F If the extension is finite and Galois, then is canonically isomorphic to.
Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.

Valuation ring

If K is a field, a valuation v is a group homomorphism from the multiplicative group K* to a totally ordered abelian group G such that, for any f, g in K with f + g nonzero, v ≥ min. The valuation ring of v is the subring of K consisting of zero and all nonzero f such that v ≥ 0.
Examples:
See also: Novikov ring and uniserial ring.

Rings with extra structure

A ring may be viewed as an abelian group, 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:
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

Cohomology ring of a topological space

To any topological space X one can associate its integral cohomology ring
a graded ring. There are also homology groups 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 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 natural product, which is analogous to the observation that one can multiply pointwise a k-multilinear form and an l-multilinear form to get a -multilinear form.
The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.

Burnside ring of a group

To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.

Representation ring of a group ring

To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.

Function field of an irreducible algebraic variety

To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry 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 in Ab, the category of abelian groups. The monoid action of a ring R on an abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space – 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 be its endomorphism ring. Note that, essentially, End is the set of all morphisms of A, where if f is in End, and g is in End, the following rules may be used to compute f + g and f · g:
where + as in f + g is addition in A, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring,, is an abelian group. Furthermore, for every r in R, right multiplication by r gives rise to a morphism of, by right distributivity. Let A =. Consider those endomorphisms of A, that "factor through" right multiplication of R. In other words, let EndR be the set of all morphisms m of A, having the property that m = r · m. 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 EndR, is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group. In essence, the most general form of a ring, is the endomorphism group of some abelian X-group.
Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

Generalization

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

Rng

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

Nonassociative ring

A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.

Semiring

A semiring is obtained by weakening the assumption that is an abelian group to the assumption that is a commutative monoid, and adding the axiom that 0 · a = a · 0 = 0 for all a in R.
Examples:

Ring object in a category

Let C be a category with finite products. Let pt denote a terminal object of C. A ring object in C is an object R equipped with morphisms , , , , and satisfying the usual ring axioms. Equivalently, a ring object is an object R equipped with a factorization of its functor of points through the category of rings:.

Ring scheme

In algebraic geometry, a ring scheme over a base 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.

Ring spectrum

In algebraic topology, a ring spectrum is a spectrum X together with a multiplication and a unit map from the sphere spectrum S, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.

Citations

General references