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:
# $r (m + n) = rm + rn$
# $(r + s) m = rm + sm$
# $(rs)m = r(sm)$
# $1m = m$
# $0_R m = r 0_M = 0_M$.

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 $\mathbb$-semimodule in the same way that an abelian group is a $\mathbb$-module.

References

Algebraic structures
Module theory

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