Generalized Skew Derivations with Engel Conditions on Lie Ideals

Let R be a prime ring and L a noncommutative Lie ideal of R. Suppose that f is a nonzero right generalized β-derivation of R associated with a β-derivation δ such that [f(x), x]k = 0 for all x ∈ L, where k is a fixed positive integer. Then either there exists s ∈ C scuh that f(x) = sx for all x ∈ R or R ⊆ M2(F ) for some field F . Moreover, if the latter case holds, then either charR = 2 or charR 6= 2 and f(x) = bx − xc for all x ∈ R, where b, c ∈ FR and b+ c ∈ C. Recently, M. C. Chou and C. K. Liu [5] proved that if δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R such that [δ(x), x]k = 0 for all x ∈ L, where k is a fixed positive integer, then charR = 2 and R ⊆ M2(F ) for some field F . This result generalizes some known results, see for instances, [15] and [20]. In this paper we extend [5] further to the so-called right generalized skew derivations. Notice that our result also generalizes the case of generalized derivations by N. Argac, L. Carini and V. De Fillipis [1]. Throughout this paper, R is always a prime ring with center Z. For x, y ∈ R, set [x, y]1 = [x, y] = xy − yx and [x, y]k = [[x, y]k−1, y] for k > 1. Notice that an Engel condition is a polynomial [x, y]k = ∑k i=0(−1) i ( k i ) yixyk−i in noncommutative indeterminantes x and y. For two subsets A and B of R, [A,B] is defined to be the additive subgroup of R generated by all elements Received February 16, 2011 and in revised form July 26, 2011. AMS Subject Classification: 16W20, 16W25, 16W55.