On the Drazin index of regular elements
type: Item
instance of: scholarly article
title: On the Drazin index of regular elements
zbMATH Open document ID: 1188.15005
author: P. Patrício
author: António Veloso da Costa
published in: Central European Journal of Mathematics
publication date: 2009-11-24T00:00:00Z
full work available at URL: http://hdl.handle.net/1822/10534
review text: Let \(R\) denote a ring with unity 1. An element \(a\in R\) is regular if \(a\in aRa\), and if some power of \(a\) is regular, then \(a\) is weak-regular. An element \(a\) is Drazin invertible if the three equations \(xax= x\), \(ax= xa\), \(a^kxa= a^k\) have a common solution \(x= a^D\) for some integer \(k\geq 0\). It is well-known that if \(a^D\) exists, it is unique. The minimum \(k\) fulfilling the above is the Drazin index \(i(a)\) of \(a\). The authors relate the Drazin index of \(a\) to the Drazin index of \(a^2 a^-+ 1- aa^-\). They also give an alternative characterization when considering matrices over an algebraically closed field, and close with some questions and remarks.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 5637474
zbMATH Keywords: Drazin inverse
zbMATH Keywords: Drazin index
zbMATH Keywords: Dedekind finite ring
zbMATH Keywords: regular ring
reviewed by: Rabe von Randow
MaRDI profile type: Publication
OpenAlex ID: W2088262917
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DOI: 10.2478/S11533-009-0015-6
description: scientific article; zbMATH DE number 5637474