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

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.

Further results on common properties of the products ac and bd

“…The set composed of generalized Drazin invertible elements in R will be denoted by R gD . In [18], the authors obtained the following analogue of Jacobson's lemma for generalized Drazin inverse under the assumption (1.3), which gives an affirmative answer to a conjecture of [15].…”

“…Corach et al in [7] generalized (1.1) and many of its relatives to the case that (1.2) aba = aca, see also [20,21,22,23]. Recently, it has been realized that there are proper counterparts of Jacobson's lemma for Drazin inverse and generalized Drazin inverse under the new condition (1.3) acd = dbd, dba = aca, see [15,18]. Obviously, the case "a = d" in (1.3) gives (1.2), the case "b = c" in (1.2) results in aca = aca.…”

“…Obviously, the case "a = d" in (1.3) gives (1.2), the case "b = c" in (1.2) results in aca = aca. This paper is a continuation of [15,18]. In the presence of (1.3), common properties of the products ac and bd are further studied in various categories.…”

On Strongly pi-Regular Rings with Involution

“…According to [24], a ∈ R is p-Drazin invertible exactly when a is a pseudo-polar element (i.e., there exists p 2 = p ∈ comm 2 (a) such that a+p ∈ U(R) and ap ∈ J(R)). For more interesting properties of p-Drazin inverses, one may refer to [11,19,25,26] and the reference therewith.…”