m-weak group inverse in a ring with proper involution

TL;DR

This paper explores new properties of the m-weak group inverse in rings with involution, introduces a decomposition, and relates it to other inverses, extending understanding of generalized inverses in algebraic structures.

Contribution

It introduces the m-weak group decomposition, characterizes the inverse via a polar-like property, and links it to the core-EP inverse, advancing the theory of generalized inverses.

Findings

Established the equivalence of m-weak group decomposition and invertibility.

Provided a polar-like characterization of the m-weak group inverse.

Connected the m-weak group inverse with the core-EP inverse for matrices and rings.

Abstract

The m-weak group inverse was recently studied in the literature. The purpose of this paper is to investigate new properties of this generalized inverse for ring elements. We introduce the m-weak group decomposition for a ring element and prove that it coincides with its m-weak group invertibility. We present the equivalent characterization of the m-weak group inverse by using a polar-like property. The relations between m-weak group inverse and core-EP inverse are also established. These give some new properties of the weak group inverse for complex matrices and ring elements.