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 $$R \times M$$ to M satisfying the following axioms:
 * 1) $$r (m + n) = rm + rn$$
 * 2) $$(r + s) m = rm + sm$$
 * 3) $$(rs)m = r(sm)$$
 * 4) $$1m = m$$
 * 5) $$0_R m = r 0_M = 0_M$$.

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 $$m \in M$$, 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 $$\mathbb{N}$$-semimodule in the same way that an abelian group is a $$\mathbb{Z}$$-module.