Commuting and centralizing generalized derivations on Lie ideals in prime rings.
type: Item
instance of: scholarly article
title: Commuting and centralizing generalized derivations on Lie ideals in prime rings.
zbMATH Open document ID: 1241.16027
DOI: 10.1134/S0001434610110143
author: Vincenzo De Filippis
author: Francesco Rania
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \(R\) be a noncommutative prime ring with \(\text{char\,}R\neq 2\), Utumi quotient ring \(U\), extended centroid \(C\), and noncentral Lie ideal \(L\). For \(x,y\in R\) set \([x,y]_1=xy-yx\) and for \(k\geq 1\) let \([x,y]_{k+1}=[[x,y]_k,y]_1\). The main result assumes that \(F\) and \(G\) are generalized derivations of \(R\) so that for a fixed integer \(k\geq 1\) and all \(x\in L\), \([F(x),x]_kx=x[G(x),x]_k\). The authors prove that for all \(x\in R\), \(F(x)=xa\) and \(G(x)=(a+z)x\) for some \(a\in U\) and \(z\in C\), or else \(R\) satisfies the standard identity \(S_4\) and one of the following holds: a) either \(F\) and \(G\) are inner with closely related constants, or b) for all \(x\in R\), \(F(x)=ax+d(x)\) and \(G(x)=bx+cx-d(x)\) for \(a,b,c\in U\) with \(a+b-c\in C\) and some derivation \(d\) of \(U\). When \(L=R\) the first conclusion, \(F(x)=ax\) and \(G(x)=(a+z)x\), must hold.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R60
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 5980718
zbMATH Keywords: generalized derivations
zbMATH Keywords: prime rings
zbMATH Keywords: functional identities
zbMATH Keywords: Lie ideals
MaRDI profile type: Publication
cites work: Generalized derivations of left faithful rings
cites work: Generalized derivations in rings*
cites work: Derivations in Prime Rings
cites work: Prime rings satisfying a generalized polynomial identity
cites work: Commuting maps: a survey.
cites work: Differential identities, Lie ideals, and Posner's theorems
cites work: Commuting maps on lie ideals
cites work: An Engel Condition with Derivation
cites work: An Engel condition with generalized derivations on Lie ideals.
cites work: Q4865491
cites work: Q4024510
cites work: Q4186484
cites work: Gpis Having Coefficients in Utumi Quotient Rings
cites work: Lie ideals and derivations of prime rings
cites work: Prime nonassociative algebras
cites work: On a new proof of Litoff's theorem
cites work: Q4714797
cites work: Q5639810
cites work: Q3033935
cites work: Lie structure of prime rings of characteristic 2
full work available at URL: s0001434610110143
OpenAlex ID: W2079091651
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description: scientific article; zbMATH DE number 5980718