Definition:Additive Semiring

Definition
An additive semiring is a semiring with a commutative distributand.

That is, an additive semiring is a ringoid $\left({S, *, \circ}\right)$ in which:
 * $(1): \quad \left({S, *}\right)$ forms a commutative semigroup
 * $(2): \quad \left({S, \circ}\right)$ forms a semigroup.

That is, such that $\left({S, *, \circ}\right)$ has the following properties:

These are called the additive semiring axioms.

Note on Terminology
The term additive semiring was coined by ProofWiki to describe this structure.

Most of the literature simply calls this a semiring; however, on ProofWiki the term semiring is reserved for more general structures, not imposing that the distributand be commutative.

Also see

 * Ringoid
 * Semiring
 * Rig
 * Ring