It has been suggested that Noncommutative ring be merged into this article. (Discuss) Proposed since February 2013.

Chapter IX of David Hilbert's Die Theorie der algebraischen Zahlkörper. The chapter title is Die Zahlringe des Körpers, literally "the number rings of the fields". The word "ring" is the contraction of "Zahlring".

In mathematics, and more specifically in abstract algebra, a ring is an algebraic structure that abstracts and generalizes the basic arithmetic operations, specifically the operations of addition and multiplication. Rings are specifically studied in the branch of mathematics known as algebra, but are used in most areas of mathematics, including geometry and mathematical analysis. They allow mathematicians to apply theorems in elementary algebra to non-numerical objects like polynomials, series and functions. The formal definition of rings is relatively recent (end of 19th century), and is an example of the tendency of modern mathematics to introduce, study, and manipulate abstract structures.

Briefly, a ring is an abelian group with a second binary operation that is associative and distributive over the abelian group operation. The abelian group operation is called "addition" and the second binary operation is called "multiplication" in analogy with the integers. One familiar example of a ring is the set of integers. The integers are a commutative ring, since a times b is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field (such as the real numbers) is a commutative ring in which one can do "division" by any nonzero element.

The ring theory consists of two main parts: the commutative ring theory, commonly known as commutative algebra, and the noncommutative ring theory. The former primarily concerns itself with problems and naturally-occurring ideas in algebraic number theory and algebraic geometry. Important commutative rings include fields, polynomial rings, the coordinate rings of algebraic varieties and the rings of integers of number fields. On the other hand, the noncommutative theory is based on very different methods and, therefore, may not be viewed as a generalization of the commutative case. The major part of it is the structure theory: how a ring breaks up into simple pieces such as matrix rings. Many noncommutative rings come from analysis; e.g., operator algebras and rings of differential operators. Via group rings, the theory also found applications to the representation theory of groups. In geometry, the cohomology ring of a topological space constitutes an important geometric invariant of the space.

Definition and illustration [

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, . . .

There are familiar properties for multiplication and addition of the integers. These properties serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · called addition and multiplication, that map every pair of elements of R to a unique element of R. These operations must satisfy the following properties called ring axioms (the symbol ⋅ is often omitted and multiplication is just denoted by juxtaposition.), which must be true for all a, b, c in R:

• Addition is abelian, meaning:

1. (a + b) + c = a + (b + c) (+ is associative)

2. There is an element 0 in R such that 0 + a = a (0 is the zero element)

3. a + b = b + a (+ is commutative)

4. For each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)

• Multiplication ⋅ is associative:

5. (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)

• Multiplication distributes over addition:

6. a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)

7. (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c).

For many authors, these seven axioms are all that are required in the definition of a ring (such is structure is also called pseudo-ring, or a rng). For others, the following additional axiom is also required:

• Multiplicative identity

8. There is an element 1 in R such that a ⋅ 1 = 1 ⋅a = a

Rings which satisfy all eight of the above axioms are sometimes, for emphasis, referred to as unital rings (also called unitary rings, rings with unity, rings with identity or rings with 1).^[1] For example, the set of even integers satisfies the first seven axioms, but it does not have a multiplicative identity, and therefore does not satisfy the eighth axiom.

Regarding which convention should be used, Gardner and Wiegandt argue that if one requires all rings to have a unity, 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."^[2] This article adopts the convention that, unless otherwise stated, a ring is assumed to be unital.

Although ring addition is commutative, so that a + b = b + a, ring multiplication is not required to be commutative; a ⋅ b need not equal b ⋅ a. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.

Some basic properties of a ring follow immediately from the axioms.

• The additive identity and the additive inverse are unique.
• The binomial formula holds for any commuting elements (i.e., ).

Example: Integers modulo 4 [

Consider the set Z₄ consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows. To avoid possible confusions and to keep the usual notation for the arithmetic operations, we will over-line 0, 1, 2, 3 when considering them in Z₄.

• (Addition) The sum in Z₄ is the remainder of (as an integer) when divided by 4. For example, in Z₄ we have and

• (Multiplication) The product in Z₄ is the remainder of (as an integer) when divided by 4. For example, in Z₄ we have and

If x is an integer, the remainder of x when divided by 4 is an element of Z₄, and this element is often denoted by "x mod 4", or sometimes , which is coherent with above notation. By checking each axiom, one verifies that Z₄ is a ring under these operations. Each axiom follows from the fact that the integers form a ring, and converting the integers to Z₄. The additive inverse of any in Z₄ is the remainder In other words, we have For example, in Z₄, we have

Once one has checked that the ring axioms hold, operations within the ring Z₄ become easier to carry out. For example, to compute 3 ⋅ (3 − 1) + 1, one first computes the value within the full set of integers (which is 7), and then converts the result by finding the remainder after dividing by 4, which in this case is 3.

Example: 2 by 2 matrices [

Consider the set of 2 by 2 matrices, whose entries are real numbers. This set is written:

One can check that with the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms. The element is the multiplicative identity element of the ring. This ring is one of the simplest examples of a non-commutative ring. To see that it is not commutative, consider the following multiplications, which give two matrices A and B such that AB is different from BA:

One can generalize this construction by replacing the set of real numbers with any ring (non-necessarily commutative), and instead of using 2 by 2 matrices, one can use square matrices of any fixed size; see matrix ring.

Rings with extra structure [

A ring may be viewed as an abelian group (by using the addition operation), with extra structure. 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 K. For instance, the set of n by n matrices over the real field R has dimension n² as a real vector space.
• A ring R is a topological ring if its set of elements is given a topology which makes the addition map ( ) and the multiplication map ( ) to be both continuous as maps between topological spaces (where X 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.

History [

Richard Dedekind, one of the founders of ring theory.

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-19th century undercut the pre-eminence of fields in mathematical analysis.

In the 1880s Richard Dedekind introduced the concept of a ring,^[3] 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. 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),^[4] 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 specific ring that had the property of "circling directly back" to an element of itself.^[5] 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.

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.^[3][6] In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.^[3]

Basic concepts [

Elements in a ring [

A left zero-divisor is an element of a ring such that there exists a non-zero element of such that .^[7] A right zero-divisor is defined similarly.

A nilpotent element is an element such that for some . One example of a nilpotent element is a nilpotent matrix. A nilpotent element is necessarily a zero-divisor.

An idempotent is an element such that . One example of an idempotent element is a projection in linear algebra.

When an element has a multiplicative inverse, it is unique, is denoted by and is called a unit. The set of units is a group under ring multiplication; this group is denoted by or . For example, if R is the set of all square matrices of size n, consists of the set of all invertible matrices of size n; called the general linear group.

Subring [

A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S. Equivalently, S is a subring if for any x, y in S, , and are in S. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R.^[8] So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring.

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[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.

The 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 (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, is never zero for any positive integer n, and those rings are said to have characteristic zero.

Given a ring R, let denote the set of all elements x in R such that x commutes with every element in R: for any y in R. Then is a subring of R; called the center of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the centralizer (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they generate a subring of the center.

Ideal [

The definition of an ideal in a ring is analogous to that of normal subgroup in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, the name "ideal". Like elements of rings, the study of ideals is central to structural understanding of a ring.

Let R be a ring. A subset I of R is then said to be a left ideal in R if . Here, denotes the span of I over R; i.e., the set of finite sums

Similarly, I is said to be right ideal if . A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E. Similarly, the two-sided ideal generated by E is the smallest two-sided ideal containing E, or, equivalently,

Given right (or left, or two-sided) ideals A and B of R, it is possible to show that the set intersection of A with B is an ideal of the same type as A and B. It is also possible to define a product of ideals such that AB is another ideal of the same sidedness as A and B. The sum of ideals A+B is also an ideal of the same type as A and B.

If x is in R, then and are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal is written as . For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.

Like a group, a ring is said to be a 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 we have that implies either or This later formulation illustrates the idea of ideals as generalizations of elements. Prime ideals are vital to the theory of commutative algebra and for algebraic geometry.

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:

• f(a + b) = f(a) ‡ f(b)

• f(a · b) = f(a) * f(b)

• f(1_R) = 1_S

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 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. The Galois group of a field extension is the set of all automorphisms of L whose restrictions to K are the identity.

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 quotient ring Z/4Z ("quotient ring" is defined below). For another example, let be a unit in a ring R. Then is a ring homomorphism, called an inner automorphism of R. For another example, let R be a commutative ring of prime characteristic p. Then is a ring endmorphism of R called the Frobenius homomorphism.

Given a ring homomorphism , the set of all elements mapped to 0 by f is called the kernel of f. It is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.

To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A. Indeed, if f is such a map, one can define the scalar multiplication by R on A as . Conversely, if A is an algebra over R, then is such a ring homomorphism.

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 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.

Like the case of a quotient group, there is a canonical map given by ; it is surjrctive. It satisfies the universal property: if is a ring homomorphism such that , then there is a unique such that . In particular, taking I to be the kernel, one sees that the quotient ring is isomorphic to the image of f; the fact known as the first isomorphism theorem. The last fact implies that actually any surjective ring homomotphism satisfies the universal property since the image of such a map is a quotient ring.

A subset of R and the quotient are related in the following way. A subset of R is called a system of representatives of if no two elements in the set belong to the same coset; i.e., each element in the set represents a unique coset. It is said to be complete if the restriction of to it is surjective.

Basic examples [

• The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
• The rational, real and complex numbers form commutative rings (in fact, they are even fields).
• The Gaussian integers form a ring, as do the Eisenstein integers. So does their generalization Kummer ring. cf. quadratic integers.
• In general, the set of all algebraic integers forms a ring. This follows for example from the fact that it is the integral closure of the ring of rational integers in the field of complex numbers. The rings in the previous example are subrings of this ring.
• The polynomial ring R[X] of polynomials over a ring R is also a ring.
• The set of formal power series R[[X₁, …, X_n]] over a commutative ring R is a ring.
• 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 corresponds to a ring of sets and is an example of a Boolean ring.
• The set of all continuous real-valued functions defined on thee real line forms a commutative ring. The operations are addition and multiplication of functions.
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