Generalized Cline's formula and Jacobson's lemma in a ring

TL;DR

This paper extends classical formulas related to the g-Drazin inverse in rings, providing generalized versions of Cline's formula and Jacobson's lemma that encompass and extend previous results in algebraic structures.

Contribution

It introduces new generalized formulas for the g-Drazin inverse in rings, broadening the scope of existing algebraic inverse results.

Findings

Generalized Cline's formula for g-Drazin inverse

Generalized Jacobson's lemma for g-Drazin inverse

Extension of known results in ring theory

Abstract

We present new generalized Cline's formula and Jacobson's lemma for the g-Drazin inverse in a ring. These extend many known results, e.g., Chen and Abdolyousefi (Generalized Jacobson's Lemma in a Banach algebra, Comm. Algebra, {\bf 49}(2021), 3263--3272), Yan and Zeng (The generalized inverses of the products of two elements in a ring, Turk. J. Math., {\bf 44}(2020), 1744--1756).