Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

Ren, Yanxun; Jiang, Lining

doi:10.3336/gm.57.1.01

Cited by 3 publications

(1 citation statement)

References 35 publications

(35 reference statements)

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“…We now have the following statement for pns-Drazin inverses which somewhat improves the corresponding results from [14] and [22]. Lemma 3.9.…”

Section: Pns-drazin Inverses With Involutionsupporting

confidence: 63%

On Strongly pi-Regular Rings with Involution

Cui¹,

Danchev²

2022

Communications in Mathematics

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Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n, depending on a, such that a^n belongs to the intersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study of the notion of a strongly pi-star-regular ring, which is the star-version of strongly pi-regular rings and which was originally introduced by Cui-Wang in J. Korean Math. Soc. (2015). We also establish various properties of these rings and give several new characterizations in terms of (strong) pi-regularity and involution. Our results also considerably extend recent ones in the subject due to Cui-Yin in Algebra Colloq. (2018) proved for pi-star-regular rings and due to Cui-Danchev in J. Algebra Appl. (2020) proved for star-periodic rings.

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“…We now have the following statement for pns-Drazin inverses which somewhat improves the corresponding results from [14] and [22]. Lemma 3.9.…”

Section: Pns-drazin Inverses With Involutionsupporting

confidence: 63%

On Strongly pi-Regular Rings with Involution

Cui¹,

Danchev²

2022

Communications in Mathematics

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Left and right-Drazin inverses in rings and operator algebras

Ren

Jiang

2022

J. Algebra Appl.

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The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

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Extension of the generalized n-strong Drazin inverse

Mosic,

Zou,

Wang

2023

Filomat

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The aim of this paper is to present an extension of the generalized n-strong Drazin inverse for Banach algebra elements using a g-Drazin invertible element rather than a quasinilpotent element in the definition of the generalized n-strong Drazin inverse. Thus, we introduce a new class of generalized inverses which is a wider class than the classes of the generalized n-strong Drazin inverse and the extended generalized strong Drazin inverses. We prove a number of characterizations for this new inverse and some of them are based on idempotents and tripotents. Several generalizations of Cline?s formula are investigated for the extension of the generalized n-strong Drazin inverse.

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Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

Cited by 3 publications

References 35 publications

On Strongly pi-Regular Rings with Involution

On Strongly pi-Regular Rings with Involution

Left and right-Drazin inverses in rings and operator algebras

Extension of the generalized n-strong Drazin inverse

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