Ring

A ring is a set $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :

1. $R$ is an 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.