On the (b, c)–Inverse in Rings

Abstract

We present new characterizations for the existence of the (b, c)–inverse in a ring. The set of all (b, c)–invertible elements is described too. Necessary and sufficient conditions which ensure that the (b, c)–inverse of a given element commutes with that element are investigated. As an application of these results, we obtain new characterizations for the existence of the image-kernel (p, q)–inverse.