Generalized Zhou inverses in rings

TL;DR

This paper introduces a new class of generalized inverses called Zhou inverses in rings, providing characterizations, properties, and formulas such as Cline's and Jacobson's for these inverses.

Contribution

It defines the generalized Zhou inverse in rings and establishes key properties, characterizations, and formulas, extending the theory of generalized inverses.

Findings

Characterization of generalized Zhou inverses via idempotents and Jacobson radical

Cline's formula for generalized Zhou inverses

Characterization of Zhou inverse in rings

Abstract

We introduce and study a new class of generalized inverses in rings. An element $a$ in a ring $R$ has generalized Zhou inverse if there exists $b\in R$ such that $bab=b, b\in comm^2(a), a^n-ab\in \sqrt{J(R)}$ for some $n\in {\Bbb N}$. We prove that $a\in R$ has generalized Zhou inverse if and only if there exists $p=p^2\in comm^2(a)$ such that $a^n-p\in \sqrt{J(R)}$ for some $n\in {\Bbb N}$. Cline's formula and Jacobson's Lemma for generalized Zhou inverses are established. In particular, the Zhou inverse in a ring is characterized.