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