# Ring

[https://en.wikipedia.org/wiki/Ring\_(mathematics](https://en.wikipedia.org/wiki/Ring_\(mathematics))

A ring consists of a set equipped with 2 binary operations that generalize addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

A **ring** is an _abelian group_ with a second binary operation that is associative, distributive over the abelian group operation, and has the identity element (although the identity is not required by some authors).

By extension from the integers, the abelian group operation is called _addition_ and the second binary operation is called _multiplication_.

Commutativity of a ring's second operation has profound impact on its behaviour, promoting _commutative ring theory_ (commutative algebra) as a key topic in ring theory.

Examples of _commutative rings_:

* the set of integers equipped with the addition and multiplication
* the set of polynomials equipped with their addition and multiplication
* the coordinate ring of an affine algebraic variety
* the ring of integers of a number field

Examples of _noncommutative rings_:

* 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
* the cohomology ring of a topological space in topology
