An Engel condition with b -generalized derivations for Lie ideals

Pao Kuei Liau, Cheng-Kai Liu

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]

Original languageEnglish
Article number1850046
JournalJournal of Algebra and its Applications
Volume17
Issue number3
DOIs
Publication statusPublished - 2018 Mar 1

Fingerprint

Skew Derivation
Lie Ideal
Generalized Derivation
Algebra
Extended Centroid
Matrix Ring
Prime Ring
Generalise
Integer
Theorem

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Applied Mathematics

Cite this

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An Engel condition with b -generalized derivations for Lie ideals. / Liau, Pao Kuei; Liu, Cheng-Kai.

In: Journal of Algebra and its Applications, Vol. 17, No. 3, 1850046, 01.03.2018.

Research output: Contribution to journalArticle

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AB - Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]

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