Homology with respect to a kernel transformation

In this article we first give the relations between commonly used images of a morphism in a category. We then investigate d-homology in a category with certain properties, for a kernel transformation d.In particular, we show that, in an abelian category, d-homology, where d is induced by the subtraction operation, is the standard homology and that in more general categories the d-homology for a trivial d is zero. We also compute through examples the d-homology for certain kernel transformations d in such categories as R-modules, abelian groups and short exact sequences of R-modules. Finally, we characterize kernel transformations in the categories of R-modules, finitely generated R-modules, partial sets and pointed sets.