Generalized Jacobson’s lemma for generalized Drazin inverses

We present new generalized Jacobson?s lemma for generalized Drazin inverses. This extends the main results on g-Drazin inverse of Yan, Zeng and Zhu (Linear & Multilinear Algebra, 68(2020), 81-93).

Further results on Left and Right Generalized Drazin Invertible Operators

In this paper we present some new characteristics and expressions of left and right generalized Drazin invertible bounded operators on a Banach space $X.$ An explicit formula relating the left and the right generalized Drazin inverses to spectral idempotents is provided. In addition, we give a characterization of operators in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) with equal spectral idempotents, where $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) denotes the set of all left (resp. right) generalized Drazin invertible bounded operators on $X.$ Next, we give some sufficient conditions which ensure that the product of elements of $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) remains in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$). Finally, we extend Jacobson's lemma for left and right generalized Drazin invertibility. The provided results extend certain earlier works given in the literature.

On gs-Drazin inverses in a ring

An element [Formula: see text] in a ring [Formula: see text] has a gs-Drazin inverse if there exists [Formula: see text] such that [Formula: see text]. In this paper, we extend Cline’s formula and Jacobson’s Lemma for gs-Drazin inverses. Various additive properties of gs-Drazin inverses are thereby obtained.

A Generalization of Strongly Nil Clean Rings

Generalizing the notion of strongly nil clean rings, we introduce strongly quasi-nil clean rings. Some fundamental properties and equivalent characterizations of this class of rings are provided. By means of g-Drazin inverses, Cline’s formula and Jacobson’s lemma for strongly quasi-nil clean elements are investigated.

Jacobson’s Lemma via Gröbner-Shirshov Bases

Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], if 1− ab is invertible then so is 1 − ba. Jacobson’s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.