Semimodule

In mathematics, a semimodule over a semiring R is an algebraic structure analogous to a module over a ring, with the exception that it forms only a commutative monoid with respect to its addition operation, as opposed to an abelian group.

Formally, a left R-semimodule consists of an additively-written commutative monoid M and a map from ${\displaystyle R\times M}$ to M satisfying the following axioms:

1. ${\displaystyle r(m+n)=rm+rn}$
2. ${\displaystyle (r+s)m=rm+sm}$
3. ${\displaystyle (rs)m=r(sm)}$
4. ${\displaystyle 1m=m}$
5. ${\displaystyle 0_{R}m=r0_{M}=0_{M}}$.

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.

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