Semimodule Explained

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.

Definition

Formally, a left R-semimodule consists of an additively-written commutative monoid M and a map from

R x M to M satisfying the following axioms:r (m + n) = rm + rn (r + s) m = rm + sm (rs)m = r(sm) 1 m = m 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 N -semimodule in the same way that an abelian group is a Z -module