Semimodule

Definition

Formally, a left R-semimodule consists of an additively-written commutative monoid M and a map from R × M to M satisfying the following axioms:

1. r ( m + n ) = r m + r n
2. ( r + s ) m = r m + s m
3. ( r s ) m = r ( s m )
4. 1 m = 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 ∈ 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 N -semimodule in the same way that an abelian group is a Z -module.