I would like to identify a representation of the subcategory of a comma category of semi-rings, whose objects are abelian group objects. When attempting to identify the representation, the following problem arise.
Let $R$ be a semi-ring and let $M$ and $N$ be $R$-semi-modules. For an $R$-semi-module homomorphism $f\colon\, M\rightarrow N$, what is a necessary and sufficient condition that $\ker (f)$ is a partitioning sub-semi-module of $M$?
I know that a sufficient condition is that $f$ is maximal.