A semiring is a set with two operations, and , such that makes into a commutative monoid, makes into a monoid, the operation distributes over , and for any , . Usually, is instead written .

A ring , can be described as a semiring for which is required to be a group. Thus every ring is a semiring. The natural numbers form a semiring, but not a ring, with the usual multiplication and addition.

Every semiring has a quasiorder given by if and only if there exists some such that . Any element with an additive inverse is smaller than any other element. Thus if has a nonzero element with an additive inverse, then the elements , 0, form a cycle with respect to . If is an idempotent operation, then is a partial order. Addition and (left and right) multiplication are order-preserving operators.