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.