One-sided $(b, c)$-inverses in rings

Abstract

In this paper we introduce a new generalized inverse in a ring -one-sided (b, c)-inverse, derived as an extension of (b, c)-inverse. This inverse also generalizes onesided inverse along an element, which was recently introduced by H. H. Zhu et al. [H. H. Zhu, J. L. Chen, P. Patrício, Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra, 64 (3) (2016) 393-403]. Also, here we present one-sided annihilator (b, c)-inverse, which is an extension of the annihilator (b, c)-inverse. Necessary and sufficient conditions for the existence of these new generalized inverses are obtained. Furthermore, we investigate conditions for the existence of one-sided (b, c)inverse of a product of three elements and we consider some properties of one-sided (b, c)inverses.

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

“…The (b, c)-inverse is a generalization of the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse. Many existence criteria and properties of the (b, c)-inverse can be found in [1,2,4,5,9,10,14,16,17,18] etc. In [4, Definition 6.2 and 6.3], Drazin introduced the hybrid (b, c)-inverse and the annihilator (b, c)-inverse of a.…”