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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.
Contents
- Definition and illustration
- Definition
- Notes on the definition
- Basic properties
- Example Integers modulo 4
- Example 2 by 2 matrices
- Dedekind
- Hilbert
- Fraenkel and Noether
- Multiplicative identity mandatory vs optional
- Basic examples
- Elements in a ring
- Subring
- Ideal
- Homomorphism
- Quotient ring
- Module
- Direct product
- Polynomial ring
- Matrix ring and endomorphism ring
- Limits and colimits of rings
- Localization
- Completion
- Rings with generators and relations
- Domains
- Division ring
- Semisimple rings
- Central simple algebra and Brauer group
- Valuation ring
- Rings with extra structure
- Some examples of the ubiquity of rings
- Cohomology ring of a topological space
- Burnside ring of a group
- Representation ring of a group ring
- Function field of an irreducible algebraic variety
- Face ring of a simplicial complex
- Category theoretical description
- Generalization
- Rng
- Nonassociative ring
- Semiring
- Ring object in a category
- Ring scheme
- Ring spectrum
- References
The conceptualization of rings started in the 1870s and 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.
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 (i.e., whether the order in which two elements are multiplied changes the result 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 the addition and multiplication of functions, 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.
Definition and illustration
The most familiar example of a ring is the set of all integers, Z, consisting of the numbers
. . . , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . .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. R is a monoid under multiplication, meaning that:
3. Multiplication is distributive with respect to addition:
Notes on the definition
As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng (or sometimes pseudo-ring). For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring.
The operations + and ⋅ are called addition and multiplication, respectively. The multiplication symbol ⋅ is often omitted, so the juxtaposition of ring elements is interpreted as multiplication. For example, xy means x ⋅ y.
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.
The additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms.
Basic properties
Some basic properties of a ring follow immediately from the axioms:
Example: Integers modulo 4
Equip the set
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 "x mod 4" or
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
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.
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" (inspired by Ernst Kummer's notion of ideal number) 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" (number ring) 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 (e.g., spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself. 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 a3 = 4a − 1, a4 = 4a2 − a, a5 = −a2 + 16a − 4, a6 = 16a2 − 8a + 1, a7 = −8a2 + 65a − 16, 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 (commutative) 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 remain 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."
In the second category, we find authors who use the following terms:
Basic examples
Commutative rings:
Noncommutative rings:
Non-rings:
Elements in a ring
A left zero divisor of a ring
A nilpotent element is an element
An idempotent
A unit is an element
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,
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[X] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2Z does not contain the identity element 1 and thus does not qualify as a subring of Z.
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
Given a ring R, let
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,
then I is a left ideal if
If x is in R, then
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
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 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 (i.e., a ring homomorphism which is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings
Examples:
Given a ring homomorphism
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 (in particular gives a structure of A-module).
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 (R, +, · ) and a two-sided ideal I of (R, +, · ), the quotient ring (or factor ring) R/I is the set of cosets of I (with respect to the additive group of (R, +, · ); i.e. cosets with respect to (R, +)) together with the operations:
(a + I) + (b + I) = (a + b) + I and(a + I)(b + I) = (ab) + I.for every a, b in R.
Like the case of a quotient group, there is a canonical map
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 with 1, 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 multiplicatively and called multiplication. The axioms of modules are the following: for all a, b in R and all x, y in M, we have:
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 of 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.
Direct product
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:
for every 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
Let R be a commutative ring and
A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let
as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central idempotents. Assume R has the above decomposition. Then we can write
By the conditions on
An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).
Polynomial ring
Given a symbol t (called a variable) and a commutative ring R, the set of polynomials
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
If R is an integral domain, then
Let
(i.e., the substitution). If S=R[t] and x=t, then f(t)=f. Because of this, the polynomial f is often also denoted by
Example:
In other words, it is the subalgebra of
Example: let f be a polynomial in one variable; i.e., an element in a polynomial ring R. Then
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism
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
(t maps to x) where
Given a non-constant monic polynomial f in
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
There are some other related constructions. A formal power series ring
together with multiplication and addition that mimic those for convergent series. It contains
Matrix ring and endomorphism ring
Let R be a ring (not necessarily commutative). The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by Mn(R). Given a right R-module
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring:
Any ring homomorphism R → S induces Mn(R) → Mn(S); 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
The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.
A ring R and the matrix ring Mn(R) over it are Morita equivalent: the category of right modules of R is equivalent to the category of right modules over Mn(R). In particular, two-sided ideals in R correspond in one-to-one to two-sided ideals in Mn(R).
Examples:
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 filtered colimit) of Ri is the ring
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're given a family of rings
For an example of a projective limit, see #completion.
Localization
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
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
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
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
The basic example is the completion Zp of Z at the principal ideal (p) 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
Similarly, the formal power series ring
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 (i.e., free algebra over the integers) with the set X of symbols; i.e., 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
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
In the category-theoretic terms, the formation
Let A, B be algebras over a commutative ring R. Then the tensor product of R-modules
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 (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. Let V be a finite-dimensional vector space over a field k and
Letting
In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.
The following is a chain of class inclusions that describes the relationship between rings, domains and fields:
Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fieldsDivision 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. 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 (or right module) over itself; i.e., a direct sum of simple modules. A ring is called a semiprimitive ring if its Jacobson radical is zero. (The Jacobson radical is the intersection of all maximal left ideals.) 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; e.g., the ring of differential operators.
Any module over a semisimple ring is semisimple. (Proof: any free module over a semisimple ring is clearly semisimple and any module is a quotient of a free module.)
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
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
For example,
Now, if F is a field extension of k, then the base extension
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:
See also: Novikov ring and uniserial ring.
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:
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 graded ring. There are also homology groups
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 theoretical 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
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). 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 but the associativity 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 (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).
Example: a tropical semiring.
Ring object in a category
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
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 Spec Z, which for any commutative ring A returns the ring Wn(A) of p-isotypic Witt vectors of length n over A.
Ring spectrum
In algebraic topology, a ring spectrum is a spectrum X together with a multiplication