On (b,c)-inverses and (c,b)-inverses

Abstract

We extend characterizations of inverses along an element to (b, c)-inverses. It is proved that if an element a in a semigroup S is both (b, c)-invertible and (c, b)-invertible for some then are group invertible, and is the (b, c)-inverse of a, is the (c, b)-inverse of a. Moreover, we establish several criteria for the existence of (b, c)-inverses and (c, b)-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.