---
type: "concept"
title: "Ring"
slug: "ring"
language: "en"
translationGroupId: "cmr4w8qxm000hqu01fygak7om"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

A ring is a [[Set|set]] $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :
1) $R$ is an [[Abelian group|abelian group]] with $+$.
2) Multiplication is associative: for all $a,b,c\in R$, $(a\times b)\times c= a\times (b\times c)$.
3) Distributivity: for all $a,b,c\in R$,
$$a\times (b+c)=a\times b+a\times c \quad \text{and}\quad (a+b)\times c=a\times c+b\times c$$
4) Multiplicative identity: there exists an element $1$ such that, for all $a\in R$, $1\times a=a\times 1=a$.

##### Remarks
- Some authors do not include the multiplicative identity in the definition. This structure of ring without an identity is often called a rng.