Nilpotent and invertible values in semiprime rings with generalized derivations.
type: Item
instance of: scholarly article
title: Nilpotent and invertible values in semiprime rings with generalized derivations.
zbMATH Open document ID: 1230.16032
author: Asma Ali
author: Shakir Ali
author: Vincenzo De Filippis
published in: Aequationes Mathematicae
publication date: 2011-10-11T00:00:00Z
full work available at URL: https://eudml.org/doc/232939
review text: The authors consider a ring \(R\) with a nonzero right ideal \(I\), a generalized derivation \(F\) of \(R\), and for a fixed positive integer \(n\) and all \(x,y\in I\), the elements \(H(x,y)=(F(xy)-yx)^n\in R\). They prove that when \(I=R\) is semiprime and all \(H(x,y)\) are either nilpotent or invertible then \(R\) is a division ring \(D\) or \(R\cong M_2(D)\). When \(R\) is a prime ring and all \(H(x,y)=0\) then \([I,I]I=0\) and \(F(x)=ax+xb\) for \(a,b\in R\) with \((a-\alpha)I=0\) and \((b-\beta)I=0\) for \(\alpha,\beta\in C\), the extended centroid of \(R\). This latter result requires the authors to first prove that \(R\) satisfies a nontrivial generalized polynomial identity.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 5956215
zbMATH Keywords: semiprime rings
zbMATH Keywords: prime rings
zbMATH Keywords: generalized derivations
zbMATH Keywords: nilpotent values
zbMATH Keywords: invertible values
zbMATH Keywords: extended centroid
MaRDI profile type: Publication
OpenAlex ID: W1966764038
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DOI: 10.1007/S00010-010-0061-Y
description: scientific article; zbMATH DE number 5956215