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In this paper, we introduce the definition of the generalized inverse f (2) T,S, which is an outer inverse of the homomorphism f of right R−modules with prescribed image T and kernel S. Some basic properties of the generalized inverse f (2) T,S are presented. It is shown that the Drazin inverse, the group inverse and the Moore-Penrose inverse, if they exist, are all the generalized inverse f (2) T,S. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse f (1,2) T,S .

The outer inverse f(2)T,S of a homomorphism of right R-modules