Characterizations of weighted (b,c) inverse

TL;DR

This paper advances the theory of weighted (b,c)-inverses in rings by providing new characterizations, existence conditions, and exploring their relationships with other inverse types, including reverse-order laws.

Contribution

It introduces new characterizations and conditions for the existence of various weighted (b,c)-inverses and their relationships with other inverse concepts in ring theory.

Findings

Established new characterizations of weighted (b,c)-inverse

Provided necessary and sufficient conditions for existence of hybrid and annihilator (v,w)-weighted (b,c)-inverses

Explored sufficient conditions for reverse-order law of annihilator (v,w)-weighted (b,c)-inverses

Abstract

The notion of weighted $(b,c)$-inverse of an element in rings were introduced, very recently [Comm. Algebra, 48 (4) (2020): 1423-1438]. In this paper, we further elaborate on this theory by establishing a few characterizations of this inverse and their relationships with other $(v, w)$-weighted $(b,c)$-inverses. We introduce some necessary and sufficient conditions for the existence of the hybrid $(v, w)$-weighted $(b,c)$-inverse and annihilator $(v, w)$-weighted $(b,c)$-inverse of elements in rings. In addition to this, we explore a few sufficient conditions for the reverse-order law of the annihilator $(v, w)$-weighted $(b,c)$-inverses.