Ring is a Set# \(R\), together with two operations \(+\) and \(*\), which has the following properties including those in #Group and #Abelian Group:
- Closure under multiplication, which means if \(a, b \in S\), then \(ab\) is also in \(S\).
- Associativity of multiplication, which means \(\forall a, b, c \in S, a(bc) = (ab)c\).
- Distributive laws, which means \(\forall a, b, c \in S, a(b + c) = ab + ac\) and \((a + b)c = ac + bc\).
If the multiplicative operation is commutative, then it forms Communicative Ring#.
If the multiplicative operation has an identity and no zero divisors, it forms Integral Domain#.