Semiring Bimodule

Let $(R,S)$ be a pair of Semirings, viewed as Monoid Objects in the Category of Commutative Monoids, equipped with the Tensor Product of Commutative Monoids. An $(R,S)$ bimodule is then a Bimodule between Monoid Objects in the Category of Commutative Monoids.

Explicitly, this means that $M$ is both a Left Module over $R$ and a Right Module over $S$ (equivalently, a left module over the Opposite of $S$) such that:

• $r\cdot(x\cdot s)=(r\cdot x)\cdot s$