# What is a necessary and sufficient condition that the kernel of a semi-module homomorphism is a partitioning sub-semi-module?

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.

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What is a partitioning subsemimodule? – Benjamin Steinberg Dec 10 '13 at 2:59
Let M be a R-semimodule and N a subsemimodule of N. Then N is a partitioning subsemimodule of M if it satisfies the following conditions; (i) there exists a subset Q of M such that M = {q + N; q in Q}, (ii) if q is not q^{'} in Q, then the intersection of {q + N} and {q^{'} + N} is empty. – Takashi Dec 13 '13 at 10:00