Characterizations of the $(b, c)$-inverse in a ring

TL;DR

This paper explores the properties and characterizations of the $(b,c)$-inverse in rings, including conditions involving decompositions, annihilators, and invertibility, and examines related idempotents and order rules.

Contribution

It provides new characterizations of the $(b,c)$-inverse in rings, linking it to decompositions, annihilators, and idempotent elements, and discusses order rules.

Findings

Characterizations of the $(b,c)$-inverse via direct sum decompositions.

Conditions involving annihilators and invertible elements for the $(b,c)$-inverse.

Analysis of the reverse order rule for the $(b,c)$-inverse.

Abstract

Let $R$ be a ring and $b, c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse, in terms of the direct sum decomposition, the annihilator and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b, c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.