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 [math]\displaystyle{ R \times M }[/math] to M satisfying the following axioms:
- [math]\displaystyle{ r (m + n) = rm + rn }[/math]
- [math]\displaystyle{ (r + s) m = rm + sm }[/math]
- [math]\displaystyle{ (rs)m = r(sm) }[/math]
- [math]\displaystyle{ 1m = m }[/math]
- [math]\displaystyle{ 0_R m = r 0_M = 0_M }[/math].
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 [math]\displaystyle{ m \in M }[/math], 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 [math]\displaystyle{ \mathbb{N} }[/math]-semimodule in the same way that an abelian group is a [math]\displaystyle{ \mathbb{Z} }[/math]-module.
References
Golan, Jonathan S. (1999), "Semimodules over semirings", Semirings and their Applications (Dordrecht: Springer Netherlands): pp. 149–161, ISBN 978-90-481-5252-0, http://dx.doi.org/10.1007/978-94-015-9333-5_14, retrieved 2022-02-22
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