Semiring

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

The term rig is also used occasionally — this originated as a joke, suggesting that rigs are rings without negative elements, similar to using rng to mean a ring without a multiplicative identity.

Definition

A semiring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:

This last axiom is omitted from the definition of a ring: it follows from the other ring axioms. Here it does not, and it is necessary to state it in the definition.