On Jacobson's lemma and Cline's formula for Drazin inverses

Dijana Mosić

doi:10.33044/revuma.v61n2a05

Abstract

Under new conditions bac = bdb and cdb = cac, we present extensions of Jacobson’s lemma and Cline’s formula for the generalized Drazin inverse and pseudo Drazin inverse in a ring. Applying these results, we give Jacobson’s lemma for the Drazin inverse, group inverse, and ordinary inverse, and Cline’s formula for the Drazin inverse.

Highlights

Let R be an associative ring with unit 1
The concept of the generalized Drazin inverse in Banach algebras was introduced by Koliha [7]
We present the generalization of Jacobson’s lemma for the Drazin inverse, group inverse, and pseudo Drazin inverse in a ring in the case that bac = bdb and cdb = cac

Summary

Introduction

Let R be an associative ring with unit 1. Rd will denote the set of all generalized Drazin invertible elements of R. Jacobson’s lemma; Cline’s formula; generalized Drazin inverse; Drazin inverse; pseudo Drazin inverse; rings. In the case that a(1 − ax) ∈ Rnil instead of a(1 − ax) ∈ Rqnil in the definition of the generalized Drazin inverse, ad = aD is the Drazin inverse of a [5].

Results

Conclusion