A class of Drazin inverses in rings

Abstract

A strongly direct sum decomposition given by a strongly nil-clean endomorphism motivates us to introduce a class of new Drazin inverses, which is called strong Drazin inverse. In this paper, some basic properties of the strong Drazin inverse are obtained. We show the Cline's Formula and Jacobson's Lemma for the strong Drazin inverse. Further, some additive results for the strong Drazin inverse are presented under additional conditions that ab = 0 or ab = ba.

On then-strong Drazin invertibility in rings

“…Then, by this characterization we prove Cline's formula and Jacobson's lemma for the n -strong Drazin inverse. The results presented extend the corresponding ones of the strong Drazin inverse [19] and Hirano inverse [2].…”

“…Later, Zhuang et al [26] extended the result of [21] to the ring case. The generalized Drazin invertibility and strong Drazin invertibility of the sums under the commutative condition were also investigated [7,8,19].…”

On Hirano inverses in rings

“…Let k ≥ 2 and the assertion works for any n ≤ k . Then As is well known, a ∈ R has strongly Drazin inverse if and only if so does 1 − a ∈ R (see [8,Lemma 3.3]). However, Hirano inveses in a ring are not the case.…”

“…The Drazin inverse is an important tool in ring theory and Banach algebra. It is very useful in matrix theory and in various applications in matrices (see [8][9][10]). The purpose of this paper is to introduce and study a new subclass of Drazin inverses.…”

“…Following [8], an element a ∈ R has strongly Drazin inverse b if a − ab ∈ N (R), ab = ba and b = bab.…”