A rig $(R, +, \cdot)$ is a set $R$ together with two binary operations $+:R^2 \to R:(a, b) \mapsto a + b$ and $\cdot:R^2 \to R:(a, b) \mapsto ab$ such that both $(R, +)$ and $(R, \cdot)$ are monoids, where $\cdot$ distributes over $+$ That is if $\{a, b, c, d\} \subset R$ then $(a+b)(c+d) = ac + ad + bc +bd$ The natural numbers with ordinary addition and multiplication $(\mathbf{N}, +, \cdot)$ is a rig.

A rig $(R, +, \cdot)$ is a ring if $(R, +)$ is a group. The integers with ordinary addition and multiplication $(\mathbf{Z}, +, \cdot)$ is a ring.