Module over a Semiring

Let $R$ be a Semiring. A $R$-module is a Commutative Monoid $M$ equipped with an operation $\cdot:R\times M\to M$ such that:

• $0\cdot x=0$
• $(r+s)\cdot x=r\cdot x+s\cdot x$
• $(rs)\cdot x=r\cdot(s\cdot x)$
• $1\cdot x=x$
• $r\cdot(x+y)=r\cdot x+r\cdot y$

Equivalently, a $R$-module is a Semiring Homomorphism $R\to\mathrm{End}(M)$ from $R$ into the Endomorphism SemiringEndomorphism Semiring of $M$.

Typically, sources distinguish between left and right $R$-modules. This distinction is redundant, as we can recover right $R$-modules as modules over the Opposite Semiring $R^{mathrm{op}}$. When $R$ is a Commutative Semiring, left and right modules coincide, and we simply speak of modules.

Properties

• Every Commutative Monoid is a module over the commutative semiring of Natural Numbers, where the action $n\cdot x$ is given by an $n$-fold sum $x$.