# 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. $0_{R}m=r0_{M}=0_{M}$ .

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.