Ring (mathematics)
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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
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.
Definition
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]
- R is an abelian group under addition, meaning that:
- (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).
- a + b = b + a for all a, b in R (that is, + is commutative).
- There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is the additive identity).
- For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the additive inverse of a).
- (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is
- R is a monoid under multiplication, meaning that:
- (a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative).
- There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the multiplicative identity).[b]
- Multiplication is distributivewith 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).
In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab.
Variations 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 "
Although ring addition is
In a ring, multiplicative inverses are not required to exist. A nonzero 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".
Illustration
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 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 x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
- If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring.
- If a ring R contains the zero ring as a subring, then R itself is the zero ring.[6]
- The binomial formulaholds for any x and y satisfying xy = yx.
Example: Integers modulo 4
Equip the set with the following operations:
- The sum in is the remainder when the integer x + y is divided by 4 (as x + y is always smaller than 8, this remainder is either x + y or x + y − 4). For example, and
- The product in is the remainder when the integer xy is divided by 4. For example, and
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,
Example: 2-by-2 matrices
The set of 2-by-2
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.
History
Dedekind
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.
Hilbert
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 a3 − 4a + 1 = 0 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 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]
Multiplicative identity and the term "ring"
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:
- to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",[32] or "ring with 1".[33]
- to omit a requirement for a multiplicative identity: "rng"[34] or "pseudo-ring",[35] although the latter may be confusing because it also has other meanings.
Basic examples
Commutative rings
- The prototypical example is the ring of integers with the two operations of addition and multiplication.
- The rational, real and complex numbers are commutative rings of a type called fields.
- A unital associative algebra over a commutative ring R is itself a ring as well as an R-module. Some examples:
- The algebra R[X] of polynomials with coefficients in R.
- The algebra of formal power series with coefficients in R.
- The set of all continuous real-valued functions defined on the real line forms a commutative -algebra. The operations are pointwise addition and multiplication of functions.
- Let X be a set, and let R be a ring. Then the set of all functions from X to R forms a ring, which is commutative if R is commutative.
- The ring of quadratic integers, the integral closure of in a quadratic extension of It is a subring of the ring of allalgebraic integers.
- The ring of profinite integers the (infinite) product of the rings of p-adic integers over all prime numbers p.
- The Hecke ring, the ring generated by Hecke operators.
- If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.
Noncommutative rings
- For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative.
- If G is an left moduleover a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by EndR(V).
- The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
- If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
- The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebrasare noncommutative.
Non-rings
- The set of natural numbers with the usual operations is not a ring, since is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to 3 to get 0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
- Let R be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution: Then R is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.
Basic concepts
Products and powers
For each nonnegative integer n, given a sequence of n elements of R, one can define the product recursively: let P0 = 1 and let Pm = Pm−1am for 1 ≤ m ≤ n.
As a special case, one can define nonnegative integer powers of an element a of a ring: a0 = 1 and an = an−1a for n ≥ 1. Then am+n = aman for all m, n ≥ 0.
Elements in a ring
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
An
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.
Subring
A subset S of R is called a subring if any one of the following equivalent conditions holds:
- the addition and multiplication of R restrictto 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 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
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
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.
Homomorphism
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:
- The function that maps each integer x to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring to the quotient ring ("quotient ring" is defined below).
- If u is a unit element in a ring R, then is a ring homomorphism, called an inner automorphism of R.
- Let R be a commutative ring of prime characteristic p. Then x ↦ xp 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 n are the identity.
- For any ring R, there are a unique ring homomorphism and a unique ring homomorphism R → 0.
- An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map is an epimorphism.
- An algebra homomorphism from a k-algebra to the endomorphism algebra of a vector space over k is called a representation of the algebra.
Given a ring homomorphism f : R → S, the set of all elements mapped to 0 by f is called the
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).
Quotient ring
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:
- If f : R → S is a ring homomorphism such that f(I) = 0, then there is a unique homomorphism such that
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.
Module
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,
- M is an abelian group under addition.
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
Constructions
Direct product
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:
for all r1, r2 in R and s1, s2 in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the
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
Polynomial 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 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 be commutative rings. Given an element x of S, one can consider the ring homomorphism
(that is, the
Example: denotes the image of the homomorphism
In other words, it is the subalgebra of k[t] generated by t2 and t3.
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' (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
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].[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 kn. 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
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
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
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 fij in S = EndR(U), resulting in the ring isomorphism:
Any ring homomorphism R → S induces Mn(R) → Mn(S).[39]
Schur's lemma says that if U is a simple right R-module, then EndR(U) is a division ring.[40] If is a direct sum of mi-copies of simple R-modules then
The
A ring R and the matrix ring Mn(R) over it are
Limits and colimits of rings
Let Ri be a sequence of rings such that Ri is a subring of Ri + 1 for all i. Then the union (or
Examples of colimits:
- A polynomial ring in infinitely many variables:
- The algebraic closure of finite fields of the same characteristic
- The field of formal Laurent seriesover a field k: (it is the field of fractions of theformal power series ring)
- The function field of an algebraic variety over a field k is where the limit runs over all the coordinate rings k[U] of nonempty open subsets U (more succinctly it is the stalk of the structure sheaf at the generic point.)
Any commutative ring is the colimit of
A
For an example of a projective limit, see § Completion.
Localization
The
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
- is a bijection between the set of all prime ideals in R disjoint from S and the set of all prime ideals in [43]
- f running over elements in S with partial ordering given by divisibility.[44]
- The localization is exact: is exact over whenever is exact over R.
- Conversely, if is exact for any maximal ideal then 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 and R-modules map to -modules.)
Completion
Let R be a commutative ring, and let I be an ideal of R.
The
The basic example is the completion of at the principal ideal (p) generated by a prime number p; it is called the ring of
Similarly, the formal power series ring R[{[t]}] 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
Rings with generators and relations
The most general way to construct a ring is by specifying generators and relations. Let F be a
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
Special kinds of rings
Domains
A
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 k[t] is a unique factorization domain, q factors into powers of distinct irreducible polynomials (that is, prime elements):
Letting 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 Now, if then such a cyclic module (for pi) has a basis in which the restriction of f is represented by a
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:
- rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
Division ring
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.
The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
A
Semisimple rings
A
Examples
- A division ring is semisimple (and simple).
- For any division ring D and positive integer n, the matrix ring Mn(D) is semisimple (and simple).
- For a field k and finite group G, the group ring kG is semisimple if and only if the order of G (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 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 R, the following are equivalent:
- R is semisimple.
- R is artinian and semiprimitive.
- R is a finite direct product where each ni is a positive integer, and each Di is a division ring (Artin–Wedderburn theorem).
Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension 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 Rn.
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
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.
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(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:
- The field of formal Laurent seriesover a field k comes with the valuation v such that v(f) is the least degree of a nonzero term in f; the valuation ring of v is theformal power series ring
- More generally, given a field k and a totally ordered abelian group G, let be the set of all functions from G to k whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by convolution:It also comes with the valuation v such that v(f) is the least element in the support of f. The subring consisting of elements with finite support is called the group ring of G (which makes sense even if G is not commutative). If G is the ring of integers, then we recover the previous example (by identifying f with the series whose nth coefficient is f(n).)
Rings with extra structure
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:
- An associative algebra is a ring that is also a vector space over a field n such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of n-by-n matrices over the real field has dimension n2 as a real vector space.
- A ring R is a topological ring if its set of elements R is given a topology which makes the addition map () and the multiplication map ⋅ : R × R → R to be both 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 exterior powers:
- For example, is a λ-ring with the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.
- A total orderingthat is compatible with ring operations.
Some examples of the ubiquity of rings
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.
Cohomology ring of a topological space
To any topological space X one can associate its integral cohomology ring
a
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
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
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 (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 EndR(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 EndR(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
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.[52]
Nonassociative ring
A
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 0 ⋅ a = a ⋅ 0 = 0 for all a in R (since it no longer follows from the other axioms).
Examples:
- the non-negative integers with ordinary addition and multiplication;
- the tropical semiring.
Other ring-like objects
Ring object in a category
Let C be a category with finite
Ring scheme
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 Wn over , which for any commutative ring A returns the ring Wn(A) of p-isotypic Witt vectors of length n over A.[53]
Ring spectrum
In algebraic topology, a ring spectrum is a spectrum X together with a multiplication and a unit map S → X from the
See also
- Algebra over a commutative ring
- Categorical ring
- Category of rings
- Glossary of ring theory
- Non-associative algebra
- Ring of sets
- Semiring
- Spectrum of a ring
- Simplicial commutative ring
Special types of rings:
- Boolean ring
- Dedekind ring
- Differential ring
- Exponential ring
- Finite ring
- Lie ring
- Local 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
- ^ This means that each operation is defined and produces a unique result in R for each ordered pair of elements of R.
- ^ The existence of 1 is not assumed by some authors; here, the term rng is used if existence of a multiplicative identity is not assumed. See next subsection.
- ^ Poonen claims that "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1".
- ^ Some other authors such as Lang further require a zero divisor to be nonzero.
- centrally primitive.
Citations
- ^ Bourbaki (1989), p. 96, Ch 1, §8.1
- ^ Mac Lane & Birkhoff (1967), p. 85
- ^ a b Lang (2002), p. 83
- ^ Isaacs (1994), p. 160
- ^ "Non-associative rings and algebras". Encyclopedia of Mathematics.
- ^ Isaacs (1994), p. 161
- ^ Lam (2001), Theorem 3.1
- ^ Lang (2005), Ch V, §3.
- ^ Serre (2006), p. 3
- ^ Serre (1979), p. 158
- ^ "The development of Ring Theory".
- ^ Kleiner (1998), p. 27
- ^ Hilbert (1897)
- ^ Cohn (1980), p. 49
- ^ Fraenkel (1915), pp. 143–145
- ^ Jacobson (2009), p. 86, footnote 1
- ^ Fraenkel (1915), p. 144, axiom R8)
- ^ a b Noether (1921), p. 29
- ^ Fraenkel (1915), p. 144, axiom R7)
- ^ van der Waerden (1930)
- ^ Zariski & Samuel (1958)
- ^ Artin (2018), p. 346
- ^ Bourbaki (1989), p. 96
- ^ Eisenbud (1995), p. 11
- ^ Gallian (2006), p. 235
- ^ Hungerford (1997), p. 42
- ^ Warner (1965), p. 188
- ^ Garling (2022)
- ^ "Associative rings and algebras". Encyclopedia of Mathematics.
- ^ Gardner & Wiegandt (2003)
- ^ Poonen (2019)
- ^ Wilder (1965), p. 176
- ^ Rotman (1998), p. 7
- ^ Jacobson (2009), p. 155
- ^ Bourbaki (1989), p. 98
- ^ Cohn (2003), Theorem 4.5.1
- ^ Jacobson (2009), p. 122, Theorem 2.10
- ^ Bourbaki (1964), Ch 5. §1, Lemma 2
- ^ a b Cohn (2003), 4.4
- ^ Lang (2002), Ch. XVII. Proposition 1.1
- ^ Cohn (1995), Proposition 1.3.1
- ^ Eisenbud (1995), Exercise 2.2
- ^ Milne (2012), Proposition 6.4
- ^ Milne (2012), end of Chapter 7
- ^ Atiyah & Macdonald (1969), Theorem 10.17 and its corollaries
- ^ Cohn (1995), pg. 242
- ^ Lang (2002), Ch XIV, §2
- ^ Weibel (2013), p. 26, Ch 1, Theorem 3.8
- ^ Milne & CFT, Ch IV, §2
- ^ Serre (1950)
- ^ Jacobson (2009), p. 162, Theorem 3.2
- ^ Jacobson (2009)
- ^ Serre, p. 44
References
- Garling, D. J. H. (2022). Galois Theory and Its Algebraic Background (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-83892-4.
- Cohn, Harvey (1980), Advanced Number Theory, New York: Dover Publications, ISBN 978-0-486-64023-5
- Serre, J-P. (1950), Applications algébriques de la cohomologie des groupes, I, II, Séminaire Henri Cartan, 1950/51
- Serre (2006), Lie algebras and Lie groups (2nd ed.), Springer [corrected 5th printing]
General references
- Artin, Michael (2018). Algebra (2nd ed.). Pearson.
- Atiyah, Michael; Macdonald, Ian G. (1969). Introduction to commutative algebra. Addison–Wesley.
- Bourbaki, N. (1964). Algèbre commutative. Hermann.
- Bourbaki, N. (1989). Algebra I, Chapters 1–3. Springer.
- ISBN 978-1-85233-587-8
- MR 1322960.
- Gallian, Joseph A. (2006). Contemporary Abstract Algebra, Sixth Edition. Houghton Mifflin. ISBN 9780618514717.
- Gardner, J.W.; Wiegandt, R. (2003). Radical Theory of Rings. Chapman & Hall/CRC Pure and Applied Mathematics. ISBN 0824750330.
- ISBN 0-88385-015-X.
- Hungerford, Thomas W. (1997). Abstract Algebra: an Introduction, Second Edition. Brooks/Cole. ISBN 9780030105593.
- Jacobson, Nathan (1964). "Structure of rings". American Mathematical Society Colloquium Publications. 37 (Revised ed.).
- Jacobson, Nathan (1943). "The Theory of Rings". American Mathematical Society Mathematical Surveys. I.
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- MR 0009016
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Special references
- Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Commutative Noetherian and Krull rings, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7
- Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Dimension, multiplicity and homological methods, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2
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- Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules with K-Theory in View. Cambridge University Press.
- ISBN 9780521432177
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- ISBN 978-0-8218-4799-2.
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- Knuth, D. E. (1998). The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison–Wesley.
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- MR 0155856
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- JSTOR 48666015
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Primary sources
- S2CID 118962421.
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- Bronshtein, I. N. and Semendyayev, K. A. (2004) ISBN 3-540-43491-7.
- History of ring theory at the MacTutor Archive
- Birkhoff, Garrett; Mac Lane, Saunders (1996), A Survey of Modern Algebra (5th ed.), New York: Macmillan
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