Semimodule

In mathematics, a semimodule over a semiring "R" is like a module over a ring except that it is only a commutative monoid rather than an abelian group.
Definition.
Formally, a left "R"-semimodule consists of an additively-written commutative monoid "M" and a map from formula_1 to "M" satisfying the following axioms:
A right "R"-semimodule can be defined similarly. For modules over a ring, the last axiom follows from the others. This is not the case with semimodules.
Examples.
If "R" is a ring, then any "R"-module is an "R"-semimodule. Conversely, it follows from the second, fourth, and last axioms that (-1)"m" is an additive inverse of "m" for all formula_7, so any semimodule over a ring is in fact a module.
Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself. Every commutative monoid is uniquely an formula_8-semimodule in the same way that an abelian group is a formula_9-module.