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 ${\displaystyle R\times M}$ to M satisfying the following axioms:

1. ${\displaystyle r(m+n)=rm+rn}$
2. ${\displaystyle (r+s)m=rm+sm}$
3. ${\displaystyle (rs)m=r(sm)}$
4. ${\displaystyle 1m=m}$
5. ${\displaystyle 0_{R}m=r0_{M}=0_{M}}$.

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 ${\displaystyle m\in M}$, 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 ${\displaystyle \mathbb {N} }$-semimodule in the same way that an abelian group is a ${\displaystyle \mathbb {Z} }$-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, retrieved 2022-02-22

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This page was last edited on 12 August 2023, at 18:55