The higher order group inverse in a ring

TL;DR

This paper extends the concept of higher-order group inverses from matrices to elements of a ring, providing new properties and characterizations for ring elements expressed as sums of such inverses and nilpotent elements.

Contribution

It introduces the higher-order group inverse in a ring and characterizes when a ring element can be written as a sum of a higher-order group element and a nilpotent element.

Findings

Extended properties of higher-order group inverse to ring elements

Characterized sum representations involving higher-order group and nilpotent elements

Derived new algebraic results for ring element decompositions

Abstract

This paper introduces and studies the higher-order group inverse in a ring. We extend known properties of the higher-order group inverse from complex matrices to elements of a ring and, in the process, derive new results. We further characterize when a ring element can be expressed as the sum of a higher-order group element and a nilpotent element.