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

TL;DR

This paper introduces a new class of one-sided generalized inverses in rings, extending existing concepts and providing conditions for their existence and properties, thereby advancing the algebraic theory of inverses.

Contribution

It defines one-sided $(b, c)$-inverses and annihilator $(b, c)$-inverses, extending prior inverse concepts, and establishes their existence conditions and properties.

Findings

Defined one-sided $(b, c)$-inverse and annihilator $(b, c)$-inverse.

Derived necessary and sufficient conditions for their existence.

Explored properties and product conditions of these inverses.

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 one-sided inverse along an element, which was recently introduced by H. H. Zhu et al. [H. H. Zhu, J. L. Chen, P. Patr\'{i}cio, 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.