The one-sided inverse along an element in semigroups and rings
type: Item
instance of: scholarly article
title: The one-sided inverse along an element in semigroups and rings
zbMATH Open document ID: 1378.15002
author: Honglin Zou
author: Huihui Zhu
author: P. Patrício
author: Jianlong Chen
published in: Mediterranean Journal of Mathematics
publication date: 2017-11-09T00:00:00Z
full work available at URL: http://hdl.handle.net/1822/50142
review text: In [Linear Algebra Appl. 434, No. 8, 1836--1844 (2011; Zbl 1219.15007)], \textit{X. Mary} introduced the concept of the inverse along an element. In this paper, the authors give a new existence criterion for the one-sided inverse along a product and characterize the existence of the Moore-Penrose inverse by means of one-sided invertibility of certain elements in a ring. For a \(\ast\)-monoid \(S\) (\(\ast: S\to S\) an involution), they show that \(a\in S^{\dagger}\cap S^\#\) if and only if \((a^\ast a)^k\) is invertible along \(a\) if and only if \((aa^\ast)^k\) is invertible along \(a\) in \(S\). Here, \(k\) is an arbitray positive integer. They also prove that the inverse of \(a\) along \(aa^\ast\) coinsides with the core inverse of \(a\) under the condition \(a\in S^{\{1,4\}}\) in \(S\).
reviewed by: Andreas Arvanitoyeorgos
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 20M99
Mathematics Subject Classification ID: 16W99
zbMATH DE Number: 6804304
zbMATH Keywords: inverse along an element
zbMATH Keywords: von Neumann regularity
zbMATH Keywords: semigroups
zbMATH Keywords: rings
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: monoid
zbMATH Keywords: core inverse
MaRDI profile type: Publication
OpenAlex ID: W2758602044
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DOI: 10.1007/S00009-017-1017-4
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description: scientific article; zbMATH DE number 6804304