On expressing matrices over \(\mathbb{Z}_2\) as the sum of an idempotent and a nilpotent
type: Item
instance of: scholarly article
title: On expressing matrices over \(\mathbb{Z}_2\) as the sum of an idempotent and a nilpotent
zbMATH Open document ID: 1482.15027
author: Janez Šter
published in: Linear Algebra and its Applications
publication date: 2018-06-12T00:00:00Z
review text: An element \(a\) in a ring \(R\) is nil-clean if it is the sum of an idempotent and a nilpotent. In [\textit{S. Breaz} et al., Linear Algebra Appl. 439, No. 10, 3115--3119 (2013; Zbl 1355.16023)] it was proved that every \(n\times n\) matrix over \(\mathbb{Z}_2\) is nil-clean. The main result of the present paper extends this result, proving that every \(n\times n\) matrix over \(\mathbb{Z}_2\) is the sum of an idempotent matrix and a nilpotent matrix with index \(\leq 4\). Furthermore, a ring in which every element is nil-clean that is not strongly \(\pi\)-regular is constructed. This answers an open question of \textit{A. J. Diesl} [J. Algebra 383, 197--211 (2013; Zbl 1296.16016)].
reviewed by: Marjan Sheibani
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 16E50
zbMATH DE Number: 6883443
zbMATH Keywords: idempotent
zbMATH Keywords: nilpotent
zbMATH Keywords: nil clean
MaRDI profile type: Publication
full work available at URL: j.laa.2018.01.015
OpenAlex ID: W2793102648
cites work: Nil-clean matrix rings
cites work: Generalizing \(\pi\)-regular rings
cites work: Nil clean rings.
cites work: Nil-clean and strongly nil-clean rings.
cites work: Lifting Idempotents and Exchange Rings
cites work: Strongly clean rings and fitting's lemma
cites work: Nil-clean quadratic elements
cites work: Q4707666
DOI: 10.1016/J.LAA.2018.01.015
description: scientific article; zbMATH DE number 6883443