Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Formally, a ring is an abelian group whose operation is called addition, with a second binary operation called multiplication that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term “rng” with a missing i to refer to the more general structure that omits this last requirement; see § Notes on the definition.)

Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.

Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.

The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.

Contents
1 Definition
1.1 Notes on the definition
2 Illustration
2.1 Some properties
2.2 Example: Integers modulo 4
2.3 Example: 2-by-2 matrices
3 History
3.1 Dedekind
3.2 Hilbert
3.3 Fraenkel and Noether
3.4 Multiplicative identity and the term “ring”
4 Basic examples
4.1 Commutative rings
4.2 Noncommutative rings
4.3 Non-rings
5 Basic concepts
5.1 Products and powers
5.2 Elements in a ring
5.3 Subring
5.4 Ideal
5.5 Homomorphism
5.6 Quotient ring
6 Module
7 Constructions
7.1 Direct product
7.2 Polynomial ring
7.3 Matrix ring and endomorphism ring
7.4 Limits and colimits of rings
7.5 Localization
7.6 Completion
7.7 Rings with generators and relations
8 Special kinds of rings
8.1 Domains
8.2 Division ring
8.3 Semisimple rings
8.3.1 Examples
8.3.2 Properties
8.4 Central simple algebra and Brauer group
8.5 Valuation ring
9 Rings with extra structure
10 Some examples of the ubiquity of rings
10.1 Cohomology ring of a topological space
10.2 Burnside ring of a group
10.3 Representation ring of a group ring
10.4 Function field of an irreducible algebraic variety
10.5 Face ring of a simplicial complex
11 Category-theoretic description
12 Generalization
12.1 Rng
12.2 Nonassociative ring
12.3 Semiring
13 Other ring-like objects
13.1 Ring object in a category
13.2 Ring scheme
13.3 Ring spectrum
14 See also