Semimodule

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 $R\times M$ to M satisfying the following axioms:

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

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 $\mathbb{\mathbb{N}}$-semimodule in the same way that an abelian group is a $\mathbb{\mathbb{Z}}$-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, http://dx.doi.org/10.1007/978-94-015-9333-5_14, retrieved 2022-02-22