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.