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}
-
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}
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} }
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