Algebraic properties of the group inverse

TL;DR

This paper explores algebraic properties of the group inverse in rings with unity, extending previous results and addressing challenges in generalizing properties related to idempotents.

Contribution

It generalizes existing results on the group inverse from specific contexts to rings with unity, highlighting new difficulties with complementary idempotents.

Findings

Extended properties of the group inverse to rings with unity

Identified challenges in generalizing properties involving complementary idempotents

Connected results to previous work in a broader algebraic setting

Abstract

We investigate properties of the group inverse in rings with unit related to products and differences of idempotents, and thus we extend some results from \cite{DENG} to more general settings. We show that most part of \cite{DENG} is easily translated into the context of rings with unit, and that there are some difficulties in generalizing statements dealing with complementary idempotents.