Existence criteria and expressions of the (b,c)-inverse in rings and its applications

Abstract

Let R be a ring. Existence criteria for the (b, c)-inverse are given.We present explicit expressions for the (b, c)-inverse by using inner inverses. We answer the question when the (b, c)-inverse of a ∈ R is an inner inverse of a. As applications, we give a unified theory of some well-known results of the {1, 3}-inverse, the {1, 4}-inverse, the Moore-Penrose inverse, the group inverse and the core inverse.