  In this paper, we introduce and investigate a new notion of exact sequences
of semimodules over semirings relative to the canonical image factorization.
Several homological results are proved using the new notion of exactness
including some restricted versions of the Short Five Lemma and the Snake Lemma
opening the door for introducing and investigating homology objects in such
categories. Our results apply in particular to the variety of commutative
monoids extending results in homological varieties.
