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 language | English |
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Article number | 1850046 |
Journal | Journal of Algebra and its Applications |
Volume | 17 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2018 Mar 1 |
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Applied Mathematics