  We consider a notion of exact sequences in any -not necessarily exact-
pointed category relative to a given (E;M)-factorization structure. We apply
this notion to 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 to relative homological
varieties.
