Regular morphisms in abelian categories

We establish some properties involving regular morphisms in abelian categories. We show a decomposition theorem on the image of a regular sum of morphisms, a characterization of regular morphisms in terms of consecutive pairs of morphisms, and a description of certain equivalent morphisms. We also generalize Ehrlich's Theorem on one-sided unit regular morphisms by showing that if N is an M-regular object, then a morphism f : M -> N is left (right) unit regular if and only if there exists a split monomorphism (epimorphism) Ker(f) -> Coker(f). We also study regular morphisms determined by generalized inverses in additive categories.