Algebraic structure → Ring theory Ring theory |
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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Algebraic structures |
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Formally, a ring is a set endowed with two binary operations called addition and multiplication such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity. See § Variations on the definition.)
Whether a ring is commutative has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.
Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.
The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.
A ring is a set R equipped with two binary operations^{ [a]} + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms^{ [1]}^{ [2]}^{ [3]}
In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab.
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: /rʊŋ/) with a missing "i". For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in § History below, many authors apply the term "ring" without requiring a multiplicative identity.
Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not necessarily equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology.
In a ring, multiplicative inverses are not required to exist. A non zero commutative ring in which every nonzero element has a multiplicative inverse is called a field.
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.^{ [4]} The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab.)
There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative.^{ [5]} For these authors, every algebra is a "ring".
The most familiar example of a ring is the set of all integers consisting of the numbers
The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.
Some basic properties of a ring follow immediately from the axioms:
Equip the set with the following operations:
Then is a ring: each axiom follows from the corresponding axiom for If x is an integer, the remainder of x when divided by 4 may be considered as an element of and this element is often denoted by "x mod 4" or which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any in is For example,
The set of 2-by-2 square matrices with entries in a field F is^{ [7]}^{ [8]}^{ [9]}^{ [10]}
With the operations of matrix addition and matrix multiplication, 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, the square matrices of dimension n with entries in R form a ring; see Matrix ring.
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.^{ [11]} In 1871, Richard Dedekind defined the concept of the ring of integers of a number field.^{ [12]} In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.
The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897.^{ [13]} 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),^{[ citation needed]} 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).^{ [14]} 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 a^{3} − 4a + 1 = 0 then:
and so on; in general, a^{n} is going to be an integral linear combination of 1, a, and a^{2}.
The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915,^{ [15]}^{ [16]} but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse.^{ [17]} In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.^{ [18]}
Fraenkel's axioms for a "ring" included that of a multiplicative identity,^{ [19]} whereas Noether's did not.^{ [18]}
Most or all books on algebra^{ [20]}^{ [21]} 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,^{ [22]} Bourbaki,^{ [23]} Eisenbud,^{ [24]} and Lang.^{ [3]} There are also books published as late as 2022 that use the term without the requirement for a 1.^{ [25]}^{ [26]}^{ [27]}^{ [28]} Likewise, the Encyclopedia of Mathematics does not require unit elements in rings.^{ [29]} In a research article, the authors often specify which definition of ring they use in the beginning of that article.
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."^{ [30]} Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.^{ [c]}^{ [31]}
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:
For each nonnegative integer n, given a sequence of n elements of R, one can define the product recursively: let P_{0} = 1 and let P_{m} = P_{m−1}a_{m} for 1 ≤ m ≤ n.
As a special case, one can define nonnegative integer powers of an element a of a ring: a^{0} = 1 and a^{n} = a^{n−1}a for n ≥ 1. Then a^{m+n} = a^{m}a^{n} for all m, n ≥ 0.
A left zero divisor 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.^{ [d]} A right zero divisor is defined similarly.
A nilpotent element is an element a such that a^{n} = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.
An idempotent is an element such that e^{2} = e. One example of an idempotent element is a projection in linear algebra.
A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a^{–1}. The set of units of a ring is a group under ring multiplication; this group is denoted by R^{×} or R* or U(R). For example, if R is the ring of all square matrices of size n over a field, then R^{×} consists of the set of all invertible matrices of size n, and is called the general linear group.
A subset S of R is called a subring if any one of the following equivalent conditions holds:
For example, the ring of integers is a subring of the field of real numbers and also a subring of the ring of polynomials (in both cases, contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers does not contain the identity element 1 and thus does not qualify as a subring of one could call a subrng, however.
An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and it is called the subring generated by E.
For a ring R, the smallest subring of R is called the characteristic subring of R. It can be generated through addition of copies of 1 and −1. It is possible that n · 1 = 1 + 1 + ... + 1 (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R. In some rings, n · 1 is never zero for any positive integer n, and those rings are said to have characteristic zero.
Given a ring R, let Z(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 Z(R) is a subring of R, called the center of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the centralizer (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they (each individually) generate a subring of the center.
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
then I is a left ideal if RI ⊆ I. Similarly, a right ideal is a subset I such that IR ⊆ 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 RE is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R.
If x is in R, then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal RxR is written as (x). For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.
Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.
Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). 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 I, J we have that IJ ⊆ P implies either I ⊆ P or J ⊆ P. This latter formulation illustrates the idea of ideals as generalizations of elements.
A homomorphism from a ring (R, +, ⋅) to a ring (S, ‡, ∗) 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 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 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 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 f : R → S, the set of all elements mapped to 0 by f is called the kernel of f. The kernel is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.
To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A (which in particular gives a structure of an A-module).
The notion of quotient ring is analogous to the notion of a quotient group. Given a ring (R, +, ⋅) and a two-sided ideal I of (R, +, ⋅), view I as subgroup of (R, +); then the quotient ring R / I is the set of cosets of I together with the operations
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 : R → R / I, given by x ↦ x + I. It is surjective and satisfies the following universal property:
For any ring homomorphism f : R → S, invoking the universal property with I = ker f produces a homomorphism that gives an isomorphism from R / ker f to the image of f.
The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field ( scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring R, an R-module M is an abelian group equipped with an operation R × M → M (associating an element of M to every pair of an element of R and an element of M) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all a, b in R and all x, y in M,
When the ring is noncommutative these axioms define left modules; right modules are defined similarly by writing xa instead of ax. This is not only a change of notation, as the last axiom of right modules (that is x(ab) = (xa)b) becomes (ab)x = b(ax), if left multiplication (by ring elements) is used for a right module.
Basic examples of modules are ideals, including the ring itself.
Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.
The axioms of modules imply that (−1)x = −x, 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 f : R → S is a ring homomorphism, then S is a left module over R by the multiplication: rs = f(r)s. If R is commutative or if f(R) is contained in the center of S, the ring S is called a R- algebra. In particular, every ring is an algebra over the integers.
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:
for all r_{1}, r_{2} in R and s_{1}, s_{2} in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the 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 is a ring with componentwise addition and multiplication.
Let R be a commutative ring and be ideals such that whenever i ≠ j. 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.^{ [36]} Namely, let be rings, the inclusions with the images (in particular are rings though not subrings). Then are ideals of R and
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).
Given a symbol t (called a variable) 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 Rt is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, then Rt is a Noetherian ring. If R is a unique factorization domain, then Rt is a unique factorization domain. Finally, R is a field if and only if Rt is a principal ideal domain.
Let be commutative rings. Given an element x of S, one can consider the ring homomorphism
(that is, the substitution). If S = Rt and x = t, then f(t) = f. Because of this, the polynomial f is often also denoted by f(t). The image of the map is denoted by Rx; 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 kt generated by t^{2} and t^{3}.
Example: let f be a polynomial in one variable, that is, an element in a polynomial ring R. Then f(x + h) is an element in Rh and f(x + h) – f(x) is divisible by h in that ring. The result of substituting zero to h in (f(x + h) – f(x)) / h is f' (x), the derivative of f at x.
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism and an element x in S there exists a unique ring homomorphism such that and restricts to ϕ.^{ [37]} 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 → S. The universal property says that this map extends uniquely to
(t maps to x) 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 Rt, there exists a ring S containing R such that f is a product of linear factors in St.^{ [38]}
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 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 consists of formal power series
together with multiplication and addition that mimic those for convergent series. It contains Rt 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 (in fact, complete).
Let R be a ring (not necessarily commutative). The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by M_{n}(R). Given a right R-module U, the set of all R-linear maps from U to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of U and is denoted by End_{R}(U).
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 f_{ij} in S = End_{R}(U), resulting in the ring isomorphism:
Any ring homomorphism R → S induces M_{n}(R) → M_{n}(S).^{ [39]}
Schur's lemma says that if U is a simple right R-module, then End_{R}(U) is a division ring.^{ [40]} If is a direct sum of m_{i}-copies of simple R-modules then
The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.
A ring R and the matrix ring M_{n}(R) over it are Morita equivalent: the category of right modules of R is equivalent to the category of right modules over M_{n}(R).^{ [39]} In particular, two-sided ideals in R correspond in one-to-one to two-sided ideals in M_{n}(R).
Let R_{i} be a sequence of rings such that R_{i} is a subring of R_{i + 1} for all i. Then the union (or filtered colimit) of R_{i} is the ring defined as follows: it is the disjoint union of all R_{i}'s modulo the equivalence relation x ~ y if and only if x = y in R_{i} for sufficiently large i.
Examples of colimits:
Any commutative ring is the colimit of finitely generated subrings.
A projective limit (or a filtered limit) of rings is defined as follows. Suppose we are given a family of rings R_{i}, i running over positive integers, say, and ring homomorphisms R_{j} → R_{i}, j ≥ i such that R_{i} → R_{i} are all the identities and R_{k} → R_{j} → R_{i} is R_{k} → R_{i} whenever k ≥ j ≥ i. Then is the subring of consisting of (x_{n}) such that x_{j} maps to x_{i} under R_{j} → R_{i}, j ≥ i.
For an example of a projective limit, see § Completion.
The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring R and a subset S of R, there exists a ring 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 ^{ [41]} 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 (to be precise, )^{ [42]}
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 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. (Of course, R then maps to and R-modules map to -modules.)
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.^{ [45]} The construction is especially useful when I is a maximal ideal.
The basic example is the completion of at the principal ideal (p) generated by a prime number p; it is called the ring of p-adic integers and is denoted The completion can in this case be constructed also from the p-adic absolute value on The p-adic absolute value on is a map from to given by where denotes the exponent of p in the prime factorization of a nonzero integer n into prime numbers (we also put and ). It defines a distance function on and the completion of as a metric space is denoted by It is again a field since the field operations extend to the completion. The subring of consisting of elements x with |x|_{p} ≤ 1 is isomorphic to
Similarly, the formal power series ring R[{[t]}] is the completion of Rt 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 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.
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 → R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.^{ [46]}
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 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 (It is also the same thing as the symmetric algebra over A with symbols X.)
In the category-theoretic terms, the formation 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 is an R-algebra with multiplication characterized by
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 ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.
Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.^{ [47]} Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial q. Then, since kt is a unique factorization domain, q factors into powers of distinct irreducible polynomials (that is, prime elements):
Letting we make V a kt-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 (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 – λ_{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.^{ [48]}
The following is a chain of class inclusions that describes the relationship between rings, domains and fields:
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.
The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.
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.
The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.
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 R, the following are equivalent:
Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension 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.)
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 ^{ [49]} Since the similarity is an equivalence relation. The similarity classes A with the multiplication form an abelian group called the Brauer group of k and is denoted by Br(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, Br(k) is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). has order 2 (a special case of the theorem of Frobenius). Finally, if k is a nonarchimedean local field (for example, ), then through the invariant map.
Now, if F is a field extension of k, then the base extension induces Br(k) → Br(F). Its kernel is denoted by Br(F / k). It consists of A such that is a matrix ring over F (that is, A is split by F.) If the extension is finite and Galois, then Br(F / k) is canonically isomorphic to ^{ [50]}
Azumaya algebras generalize the notion of central simple algebras to a commutative local 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(f + g) ≥ min{v(f), v(g)}. The valuation ring of v is the subring of K consisting of zero and all nonzero f such that v(f) ≥ 0.
Examples:
A ring may be viewed as an abelian group (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:
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.
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 (k + l)-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.
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 is the set of 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.
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.
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.
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.
Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of -modules). The monoid action of a ring R on an abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".
Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f ⋅ g:
where + as in f(x) + g(x) 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, (R, +, ⋅ ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those endomorphisms of A, that "factor through" right (or left) multiplication of R. In other words, let End_{R}(A) be the set of all morphisms m of A, having the property that m(r ⋅ x) = r ⋅ m(x). It was seen that every r in R gives rise to a morphism of A: right multiplication by r. It is in fact true that this association of any element of R, to a morphism of A, as a function from R to End_{R}(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with X being its set of operators).^{ [51]} 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.
Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.
A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.^{ [52]}
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.^{[ citation needed]}
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 0 ⋅ a = a ⋅ 0 = 0 for all a in R (since it no longer follows from the other axioms).
Examples:
Let C be a category with finite products. Let pt denote a terminal object of C (an empty product). A ring object in C is an object R equipped with morphisms (addition), (multiplication), (additive identity), (additive inverse), and (multiplicative identity) 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:
In algebraic geometry, a ring scheme over a base scheme S is a ring object in the category of S-schemes. One example is the ring scheme W_{n} over , which for any commutative ring A returns the ring W_{n}(A) of p-isotypic Witt vectors of length n over A.^{ [53]}
In algebraic topology, a ring spectrum is a spectrum X together with a multiplication and a unit map S → X from the sphere spectrum S, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.
Special types of rings: