Definition:Semiring (Abstract Algebra)

Definition
A semiring is a ringoid $\left({S, *, \circ}\right)$ in which:
 * $(1): \quad \left({S, *}\right)$ forms a 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 semiring axioms.

Also defined as
There are various other conventions on what constitutes a semiring.

Some of these have a distinguished, different name on :


 * An Additive semiring is a semiring whose distributand is commutative


 * A Rig is a semiring whose distributand forms a commutative monoid

Still, some sources impose further that there be a identity element for the distributor, that is, that $\left({S, \circ}\right)$ be a monoid.

Such a structure could be referred to as a rig with unity, consistent with the definition of ring with unity.

This website thus specifically defines a semiring as one fulfilling axioms $A0, A1, M0, M1, D$ only (that is, as two semigroups bound by distributivity).

Examples

 * Definition:Semiring of Natural Numbers
 * Definition:Semiring of Ideals of Ring

Stronger properties

 * Definition:Commutative Ring with Unity
 * Definition:Commutative Ring
 * Definition:Ring (Abstract Algebra)
 * Definition:Commutative Semiring
 * Definition:Rig
 * Definition:Additive Semiring

Weaker properties

 * Definition:Ringoid