Please refer to the glossary of ring theory for the definitions of terms used throughout ring theory.

Table of contents |

2 Elementary introduction 3 Some useful theorems 4 Generalizations 5 External link |

Richard Dedekind introduced the concept of a ring.

The term *ring (Zahlring)* was coined by David Hilbert in the article *Die Theorie der algebraischen Zahlkörper,* Jahresbericht der Deutschen Mathematiker Vereiningung, Vol. 4, 1897.

The first axiomatic definition of a ring was given by A. A. Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914.

In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper *Ideal Theory in Rings*.

Formally, a ring is an abelian group (*R*, +), together with a second binary operation * such that for all *a*, *b* and *c* in *R*,

A ring is called *commutative* if its multiplication is commutative. The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers. Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Summary: Euclidean domain => principal ideal domain => unique factorization domain => integral domain => Commutative ring.

Non-commutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings. Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.