A ring is a triple \((R, +, \cdot)\) consisting of a set \(R\) and binary operations \(+\) and \(\cdot\) on \(R\) such that
\((R, +)\) is an abelian group, with identity \(0\);
\((R, \cdot)\) is a monoid, with identity \(1\);
for all \(a, b, c\in R\), \(a(b+c) = ab + ac\) and \((b+c)a = ba + ca\).
A ring is commutative if additionally \(ab = ba\) for all \(a, b\in R\).