An Engel condition with skew derivations

摘要

Let R be a prime ring and set [x, y]₁ = [x, y] = xy − yx for and inductively [x, y] _k = [[x, y] _k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] _k = 0 for all , where k is a fixed positive integer, then charR = 2 and for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne. Keywords Prime ring - Lie ideal - Automorphism - Skew derivation - Generalized polynomial identity (GPI) Mathematics Subject Classification (2000) 16W20 - 16W25 - 16W55 Communicated by D. Segal.