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 × M {\displaystyle R\times M} to M satisfying the following axioms:
r ( m + n ) = r m + r n {\displaystyle r(m+n)=rm+rn}
( r + s ) m = r m + s m {\displaystyle (r+s)m=rm+sm}
( r s ) m = r ( s m ) {\displaystyle (rs)m=r(sm)}
1 m = m {\displaystyle 1m=m}
0 R m = r 0 M = 0 M {\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 m ∈ M {\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 N {\displaystyle \mathbb {N} } -semimodule in the same way that an abelian group is a Z {\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
