Identities with Engel conditions on derivations.
type: Item
instance of: scholarly article
title: Identities with Engel conditions on derivations.
zbMATH Open document ID: 1247.16042
author: Tsiu-Kwen Lee
author: M. Tamer Koşan
author: Yiqiang Zhou
published in: Monatshefte für Mathematik
publication date: 2012-03-23T00:00:00Z
full work available at URL: 14.pdf
review text: Let \(R\) be a semiprime ring with center \(Z(R)\), symmetric Martindale quotient ring \(Q\), and nonzero derivation \(D\). The authors prove two results involving \(D\) and Engel conditions. Let \([x,y_1]_1=xy_1-y_1x\) and for \(n>1\) set \([x,y_1,\dots,y_n]_n=[[x,y_1,\dots,y_{n-1}]_{n-1},y_n]_1\). The first major result of the authors is that for fixed positive integers \(s\), \(m\), \(n_1,\dots,n_s\), if the expression \([x^m,D(x^{n_1}),\dots,D(x^{n_s})]_s=0\) for all \(x\in R\), and if \(R\) is \(k!\)-torsion free for \(k=1+\max\{m,n_j\}\), then \(D(R)\subseteq Z(R)\). The second main result replaces each \(D(x^{n_j})\) above with \(D(x)^{n_j}\) and concludes, without the \(k!\)-torsion free condition, that there is a central idempotent \(e\in Q\) so that the extension of \(D\) to \(Q\) is zero on \(eQ\) and that \((1-e)Q\) satisfies the standard polynomial identity \(S_4\). Further, for this second result, when \(R\) is prime, then it must satisfy \(S_4\) and if all the \(n_j\) are odd, then \(\text{char\,}R=2\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 6018281
zbMATH Keywords: semiprime rings
zbMATH Keywords: derivations
zbMATH Keywords: prime rings
zbMATH Keywords: differential identities
zbMATH Keywords: Engel conditions
MaRDI profile type: Publication
OpenAlex ID: W1982753883
Wikidata QID: Q57787800
cites work: Q4865491
cites work: Centralizing mappings and derivations in prime rings
cites work: Commuting maps: a survey.
cites work: Gpis Having Coefficients in Utumi Quotient Rings
cites work: Hypercentral derivations
cites work: Q4852753
cites work: Prime nonassociative algebras
cites work: Derivations with nilpotent values
cites work: Q5639810
cites work: Center-like elements in prime rings
cites work: PI-algebras. An introduction
cites work: An Engel Condition with Derivation
cites work: An Engel condition with derivation for left ideals
cites work: Q4024510
cites work: Semiprime Rings with Hypercentral Derivations
cites work: Q4869723
cites work: AN IDENTITY WITH GENERALIZED DERIVATIONS
cites work: Prime rings satisfying a generalized polynomial identity
cites work: Derivations in Prime Rings
cites work: Commuting and Centralizing Mappings in Prime Rings
DOI: 10.1007/S00605-010-0252-6
description: scientific article; zbMATH DE number 6018281