A Ring **(R, +, *)** is a set R with binary operations^{} + and * satisfying the following axioms:

**(R, +)**is an**abelian group**,**(R, *)**is a**semigroup**,- Multiplication
**distributes**over addition: a * (b+c) = a * b + a * c and (a+b) * c = a * c + b * c for all a, b, c in R.

Rings that also satisfy commutativity for multiplication are called **commutative rings**.

In some definitions the semigroup (R, *) is required to have an identity 1.