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.