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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:
\( r (m + n) = rm + rn \)
\( (r + s) m = rm + sm \)
\( (rs)m = r(sm) \)
\( 1m = m \)
\( 0_R m = r 0_M = 0_M. \)
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 \( \mathbb{N}\)-semimodule in the same way that an abelian group is a \( \mathbb{Z}\)-module.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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