Characterizations and representations of outer inverse for matrices over a ring

Abstract

In this paper, we investigate the existence and representations of the generalized inverse , , , , , of a matrix A over an associative ring, and give some characterizations and representations of these generalized inverses by using regularity. Some explicit expressions for and of a matrix A by means of -inverse are also given. Furthermore, we present the relationship between the generalized inverse and the -inverse of a matrix A over rings. And then using the regularity give the sufficient and necessary condition for the existence and the expression of -inverse of matrices over a ring. Some explicit expressions for -inverse of an element in semigroups by means of -inverse are also studied. As applications, some new results for the Mary inverse, the core inverse and dual core inverse are obtained.