The outer inverse $f_{T,S}^{(2)}$ of a homomorphism of right $R-$modules

Abstract

In this paper, we introduce the definition of the generalized inverse $f_{T,S}^{(2)}$, 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 $f_{T,S}^{(2)}$ 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 $f_{T,S}^{(2)}$. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse $f_{T,S}^{(1,2)}$.