Semimodule

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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[edit]

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

  1. .

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[edit]

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 , 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 -semimodule in the same way that an abelian group is a -module.

References[edit]