In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions; namely the set must be an Abelian group under addition and a monoid under multiplication^a[›] such that multiplication distributes over addition. While these operations are familiar from many mathematical structures, such as number systems or the integers—for example, they are also very general in the sense that they take a broad variety of mathematical objects into account. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of rings makes them a central organizing principle of contemporary mathematics. The branch of mathematics that studies rings is known as ring theory.

Rings share various structural aspects in common with the integers, and in particular, they share many number-theoretic properties. It is the fundamental theorem of arithmetic that every integer greater than one is the unique product of positive prime numbers. An abstract ring retains just enough structure from the integers to allow the notion of a prime number to be well-defined. However, the conclusion of the fundamental theorem of arithmetic is not satisfied by all rings - those rings which do support a translation of this theorem into the context of ring theory, are known as unique factorization domains. Likewise, rings which support an analogue of the Euclidean algorithm or Bézout's identity, are known as Euclidean domains, and Bézout domains, respectively. Therefore, although every ring shares many number-theoretic and algebraic properties in common with the integers, the similarity ends there for many rings. That said, it is often the case that the ring axioms are sufficient to develop a vast and extensive theory, not necessarily at all similar to that of the integers.

Although ring theory is only concerned with a single mathematical structure (rings), and therefore concerns only one structural aspect of the integers, it has far-reaching applications in mathematics. For example, algebraic geometry, the combination of ring theory with the language and problems of geometry, may be used to give a proof of Fermat's last theorem - a famous but now solved problem in mathematics. In a similar manner, many number theoretic facts may be proved using techniques of ring theory; sometimes there being no proof of a number theoretic fact without the use of ring theory. This is not to say, however, that ring theory only has applications in number theory. For example, ring theory has many important applications in general topology, and in particular, algebraic topology, where rings can serve as certain invariants of topological spaces - or, in more formal language, there exist certain functors between the category of topological spaces and the category of rings. One example being the cohomology ring, associated to any topological space. As another example of the applications of ring theory to mathematics, to any group is associated its Burnside ring which uses a ring to describe the various ways a group may act on a finite set. These examples provide only a taste of the applications of ring theory in mathematics - there are many other such applications to be discussed later in this article (see below).

Although ring theory is very much central and basic to much of mathematics, it is still a major area of research. A huge area of current research in ring theory, lies within algebraic geometry and algebraic number theory. Ring theory alone is a major area of research – the main aspect of research being the distinction between commutative rings, and non-commmutative rings. Commutative rings are fairly well understood compared to noncommutative rings, due to the fact that a much smaller class of rings are to be considered. On the other hand, many results in noncommutative ring theory may be easily transferred to commutative ring theory, while retaining the same set of hypotheses. Despite these facts, there are many important open problems in both commutative and noncommutative ring theory.

Contents 1 Definition and illustration 1.1 First example: the integers 1.2 Formal definition 1.3 Notes on the definition 1.4 Second example: the ring Z4 1.5 Third example: the trivial ring 2 Basic concepts 2.1 Subring 2.2 Homomorphism 2.3 Ideal 3 History 4 Elementary properties of rings 5 New rings from old 5.1 Quotient ring 5.2 Polynomial ring 6 Some examples of the ubiquity of rings 7 Special rings and rings with additional structure 7.1 Finite ring 7.2 Associative algebras 7.3 Lie ring 7.4 Topological ring 8 Commutative rings 8.1 Principal ideal domains 8.2 Unique factorization domains 8.3 Integral domains and fields 8.3.1 Relation to other algebraic structures 9 Noncommutative rings 10 Category theoretical description 11 See also 12 Notes 12.1 Citations 13 References 13.1 General references 13.2 Special references 13.3 Historical references

[ Definition and illustration

[ First example: the integers

Algebraic structures Magma Set S with binary operation + Semigroup Associativity of + Monoid Existence of identity element for + in S Group Existence of inverse elements for + in S Abelian group Commutativity of + Pseudo-ring Associative binary operation · Distributivity of · over + Ring Existence of identity element for · in S Commutative ring Commutativity of · Field Existence of inverse elements for · in S

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers

..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... ^[1]

together with the usual operations of addition and multiplication. These operations satisfy the following properties:

• The integers form an abelian group under addition; that is:
  • Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.
  • Existence of additive identity: For any integer a, a + 0 = 0 + a = a. Zero is called the identity element of the integers because adding 0 to any integer (in any order) returns the same integer.
  • Commutativity of addition: For any two integers a and b, a + b = b + a. So the order in which you add two integers is irrelevant.
  • Associativity of addition: For any integers, a, b and c, (a + b) + c = a + (b + c). So, adding b to a, and then adding c to this result, is the same as adding c to b, and then adding this result to a.
  • Existence of additive inverse: For any integer a, there exists an integer denoted by −a such that a + (−a) = (−a) + a = 0. The element, −a, is called the additive inverse of a because adding a to −a (in any order) returns the identity.

• The integers form a multiplicative monoid (a monoid under multiplication); that is:
  • Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.
  • Associativity of multiplication: Given any integers, a, b and c, (a · b) · c = a · (b · c). So multiplying b with a, and then multiplying c to this result, is the same as multiplying c with b, and then multiplying a to this result.
  • Existence of multiplicative identity: For any integer a, a · 1 = 1 · a = a. So multiplying any integer with 1 (in any order) gives back that integer. One is therefore called the multiplicative identity.
• Multiplication is distributive over addition : These two structures on the integers (addition and multiplication) are compatible in the sense that
  • a · (b + c) = (a · b) + (a · c), and
  • (a + b) · c = (a · c) + (b · c)

for any three integers a, b and c.

[ Formal definition

There are some differences in exactly what axioms are used to define a ring. Here one set of axioms is given, and comments on variations follow.

A ring is a set R equipped with two binary operations + : R × R → R and · : R × R → R (where × denotes the Cartesian product), called addition and multiplication. To qualify as a ring, the set and two operations, (R, +, · ), must satisfy the following requirements known as the ring axioms. ^[2]

• (R, +, · ) is required to be an abelian group under addition:

1.	Closure under addition.	For all a, b in R, the result of the operation a + b is also in R.c[›]
2.	Associativity of addition.	For all a, b and c in R, the equation (a + b) + c = a + (b + c) holds.
3.	Existence of additive identity.	There exists an element 0 in R, such that for all elements a in R, the equation 0 + a = a + 0 = a holds.
4.	Existence of additive inverse.	For each a in R, there exists an element b in R such that a + b = b + a = 0
5.	Commutativity of addition.	For all a, b in R, the equation a + b = b + a holds.

• (R, +, · ) is required to be a monoid under multiplication:

1.	Closure under multiplication.	For all a, b in R, the result of the operation a · b is also in R.c[›]
2.	Associativity of multiplication.	For all a, b, and c in R, the equation (a · b) · c = a · (b · c) holds.
3.	Existence of multiplicative identity.a[›]	There exists an element 1 in R, such that for all elements a in R, the equation 1 · a = a · 1 = a holds.

• The distributive laws:

1. For all a, b and c in R, the equation a · (b + c) = (a · b) + (a · c) holds.

2. For all a, b and c in R, the equation (a + b) · c = (a · c) + (b · c) holds.

This definition assumes that a binary operation on R is a function defined on R×R with values in R. Therefore, for any a and b in R, the addition a+b and the product a·b are elements of R.

The most familiar example of a ring is the set of all integers, Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, together with the usual operations of addition and multiplication.^[3]

Another familiar example is the set of complex numbers C, also with the usual addition and multiplication.

Another example of ring is the set of all square matrices of a fixed size, with real elements, using the matrix addition and multiplication of linear algebra. In this case, the ring elements 0 and 1 are the zero matrix (with all entries equal to 0) and the identity matrix, respectively.

Hamilton's quaternions also form a ring.

[ Notes on the definition

As with some axiomatic theories, there are often differences of usage in what axioms a ring should satisfy. Sometimes the disagreement between two definitions is minor. For instance, some authors insist that 1 ≠ 0 in a ring (in words, this means that the multiplicative identity of the ring must be different from its additive identity). In particular they don't consider the trivial ring to be a ring (see below).

A more significant disagreement is that some authors omit the existence of a multiplicative identity in a ring^{[4][5] [6]} . For instance, this would allow the even integers to form a ring with the natural operations of addition and multiplication (all ring axioms are satisfied except for the existence of a multiplicative identity). Rings that satisfy the ring axioms as given above but do not contain a multiplicative identity are sometimes called pseudo-rings. The term rng (jocular; ring without the multiplicative identity) is also used for such rings. Rings which do have multiplicative identities (and also satisfy the above axioms) are sometimes referred to unital rings, unitary rings, rings with unity, rings with identity or rings with 1.^[7] Note that one can always embed a non-unitary ring inside a unitary ring (see this for one particular construction of this embedding).

There are still other more significant differences between two particular definitions of a ring. For instance, some authors omit associativity of multiplication in the set of ring axioms; rings that are nonassociative are called nonassociative rings. In this article, all rings are assumed to satisfy the axioms as given above unless stated otherwise.

[ Second example: the ring Z₄

Consider the set Z₄ consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows (note that for any integer, x, x mod 4 is defined to be the remainder when x is divided by 4):

• For any x, y in Z₄, x + y is defined to be their sum in Z (the set of all integers) mod 4. So we can represent the additive structure of Z₄ by the left-most table as shown.

• For any x, y in Z₄, x ⋅ y is defined to be their product in Z (the set of all integers) mod 4. So we can represent the multiplicative structure of Z₄ by the right-most table as shown.

·	0	1	2	3
0	0	0	0	0
1	0	1	2	3
2	0	2	0	2
3	0	3	2	1

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

It is simple (but tedious) to verify that Z₄ is a ring under these operations. First of all, one can use the left-most table to show that Z₄ is closed under addition (any result is either 0, 1, 2 or 3). Associativity of addition in Z₄ follows from associativity of addition in the set of all integers. The additive identity is 0 as can be verified by looking at the left-most table. Given an integer x, there is always an inverse of x; this inverse is given by 4 - x as one can verify from the additive table. Therefore, Z₄ is an abelian group under addition.

Similarly, Z₄ is closed under multiplication as the right-most table shows (any result above is either 0, 1, 2 or 3). Associativity of multiplication in Z₄ follows from associativity of multiplication in the set of all integers. The multiplicative identity is 1 as can be verified by looking at the right-most table. Therefore, Z₄ is a monoid under multiplication.

Distributivity of the two operations over each other follow from distributivity of addition over multiplication (and vice-versa) in Z (the set of all integers).

Therefore, this set does indeed form a ring under the given operations of addition and multiplication.

Properties of this ring

• In general, given any two integers, x and y, if x ⋅ y = 0, then either x is 0 or y is 0. It is interesting to note that this does not hold for the ring (Z₄, +, ⋅):

2 ⋅ 2 = 0

although neither factor is 0. In general, a non-zero element a of a ring, (R, +, ⋅) is said to be a zero divisor in (R, +, ⋅), if there exists a non-zero element b of R such that a ⋅ b = 0. So in this ring, the only zero divisor is 2 (note that 0 ⋅ a = 0 for any a in a ring (R, +, ⋅) so 0 is not considered to be a zero divisor).

• A commutative ring which has no zero divisors is called an integral domain (see below). So Z, the ring of all integers (see above), is an integral domain (and therefore a ring), although Z₄ (the above example) does not form an integral domain (but is still a ring). So in general, every integral domain is a ring but not every ring is an integral domain.

[ Third example: the trivial ring

If we define on the singleton set {0}:

0 + 0 = 0

0 × 0 = 0

then one can verify that ({0}, +, ×) forms a ring known as the trivial ring. Since there can be only one result for any product or sum (0), this ring is both closed and associative for addition and multiplication, and furthermore satisfies the distributive law. The additive and multiplicative identities are both equal to 0. Similarly, the additive inverse of 0 is 0. The trivial ring is also a (rather trivial) example of a zero ring (see below).

[ Basic concepts

[ Subring

Informally, a subring of a ring is another ring that uses the "same" operations and is contained in it. More formally, suppose (R, +, ·) is a ring, and S is a subset of R such that

• for every a, b in S, a + b is in S;
• for every a, b in S, a · b is in S;
• for every a in S, the additive inverse −a of a is in S; and
• the multiplicative identity '1' of R is in S.

Let '+_S' and '·_S' denote the operations '+' and '·', restricted to S×S. Then (S, +_S, ·_S) is a subring of (R, +, ·).^[8] Since the restricted operations are completely determined by S and the original ones, the subring is often written simply as (S, +, ·).

For example, a subring of the complex number ring C is any subset of C that includes 1 and is closed under addition, multiplication, and negation, such as:

• The rational numbers Q
• The algebraic numbers A
• The real numbers R

If A is a subring of R, and B is a subset of A such that B is also a subring of R, then B is a subring of A.

[ Homomorphism

A homomorphism from a ring (R, +, ·) to a ring (S, ‡, *) is a function f from R to S that commutes with the ring operations; namely, such that, for all a, b in R the following identities hold:

• f(a + b) = f(a) ‡ f(b)

• f(a · b) = f(a) * f(b)

Moreover, the function f must take the identity element 1_R of '·' to the identity element 1_S of '*'.

For example, 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 Z to the ring Z₄.

If f is a ring homomorphism from (R, +, ·) to (S, ‡, *), the inverse image of the identity element 1_S of ‡ (that is, all elements of R that are mapped to 1_S by f) is a subring of (R, +, ·).

A ring homomorphism is said to be an isomorphism if it is both an epimorphism and a monomorphism in the category of rings.

[ Ideal

The purpose of an ideal in a ring is to somehow allow one to define the quotient ring of a ring (analogous to the quotient group of a group; see below). An ideal in a ring can therefore be thought of as a generalization of a normal subgroup in a group. More formally, let (R, +, · ) be a ring. A subset I of R is said to be a right ideal in R if:

• (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e (I, +) is a subgroup of (R, +)).

• For every x in I and r in R, x · r is in I.

A left ideal is similarly defined with the second condition being replaced. More specifically, a subset I of R is a left ideal in R if:

• (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e (I, +) is a subgroup of (R, +)).

• For every x in I and r in R, r · x is in I.

Notes

• If k is in R, then k · R is a right ideal in R, and R · k is a left ideal in R. These ideals (for any k in R) are called the principal right and left ideals generated by k.

• If every ideal in a ring (R, +, · ) is a principal ideal in (R, +, · ), (R, +, · ) is said to be a principal ideal ring.

• An ideal in a ring, (R, +, · ), is said to be a two-sided ideal if it is both a left ideal and right ideal in (R, +, · ). It is preferred to call a two-sided ideal, simply an ideal.

• If I = {0} (where 0 is the additive identity of the ring (R, +, · )), then I is an ideal known as the trivial ideal. Similarly, R is also an ideal in (R, +, · ) called the unit ideal.

Examples

• Any additive subgroup of the integers is an ideal in the integers with its natural ring structure.

• There are no non-trivial ideals in R (the ring of all real numbers) (i.e, the only ideals in R are {0} and R itself). More generally, a field cannot contain any non-trivial ideals.

• From the previous example, every field must be a principal ideal ring.

• A subset, I, of a commutative ring (R, +, · ) is a left ideal if and only if it is a right ideal. So for simplicity's sake, we refer to any ideal in a commutative ring as just an ideal.

[ History

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis.

Richard Dedekind (image to the right) introduced the concept of a ring, ^[9] and the term ring (Zahlring) was coined by David Hilbert in 1892 and published in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. According to Harvey Cohn, Hilbert used the term for a specific ring that had the property of "circling directly back" to an element of itself.^[10]

The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914. ^[9] In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.^[9]

[ Elementary properties of rings

Some properties of rings follow directly from the ring axioms through very simple proofs.

In particular, since the axioms state that (R,+) is a commutative group, all pertinent theorems of group theory apply, such as the uniqueness of the additive identity and the uniqueness of the additive inverse of a particular element. In the same way one can prove the uniqueness of the inverse of a unit in a ring.

However, rings also have specific properties that combine addition with multiplication. In any ring (R, +, ⋅):

• For any element a, the equations a ⋅ 0 = 0 ⋅ a = 0 hold.
• If 0 = 1, the ring is trivial (that is, R has only one element).
• For any element a, the equation (−1) ⋅ a = −a holds.
• For any elements a and b, the equations (−a) ⋅ b = a ⋅ (−b) = −(a ⋅ b)

[ New rings from old

[ Quotient ring

Informally, the quotient ring of a ring, is a generalization of 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 underlying 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.

[ Polynomial ring

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[ Some examples of the ubiquity of rings

It is remarkable how many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring. For instance:

• To any topological space X one can associate its integral cohomology ring , a graded ring. As the name suggests, 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)-mulilinear form. The importance of the ring structure in cohomology is hard to overstate: it 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 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.
• 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.

[ Special rings and rings with additional structure

[ Finite ring

Given a natural number m, how many distinct rings (not necessarily with unity) have m elements? When m is prime there are only two rings of order m; the additive group of each being isomorphic to the cyclic group of order m. One is the zero ring and the other is the Galois field.

As with finite groups, the complexity of the classification depends upon the complexity of the prime factorization of m. If m is the square of a prime, for instance, there are precisely eleven rings having order m. On the other hand, there can be only two groups having order m; both of which are abelian.

The theory of finite rings is more complex than that of finite abelian groups, since any finite abelian group is the additive group of at least two nonisomorphic finite rings: the direct product of copies of , and the zero ring. On the other hand, the theory of finite rings is simpler than that of not necessarily abelian finite groups. For instance, the classification of finite simple groups was one of the major breakthroughs of twentieth century mathematics, its proof spanning thousands of journal pages. On the other hand, any finite simple ring is isomorphic to thee ring of n by n matrices over a finite field of order q. This follows from two theorems of Source: this wikipedia article, under CC-BY-SA.