Annihilators of Power Values of a Right Generalized (α, Β)-derivation

Let R be a prime ring with a right generalized (α, β)derivation f and let a ∈ R. Suppose that af(x)n = 0 for all x ∈ R, where n is a fixed positive integer. Then af(x) = 0 for all x ∈R. In particular, if f is either a regular right generalized (α, β)-derivation or a nonzero generalized (α, β)-derivation, then a = 0. In [13] I. N. Herstein proved that if R is a prime ring and d is an inner derivation of R such that d(x)n = 0 for all x ∈ R and n is a fixed positive integer, then d = 0. In [11] A. Giambruno and I. N. Herstein extended this result to arbitrary derivations in semiprime rings. In [3] J. C. Chang and J. S. Lin extended this result further to (α, β)-derivation. Recently, Lee and Liu [18] and the author [5] extended this result independently further to generalized skew derivations (right generalized (α, β)-derivations). @@ In [1] M. Bres̆ar gave a generalization of the result due to I. N. Herstein and A. Giambruno [11] in another direction. Explicitly, he proved the following: Let R be a semiprime ring with a derivation d, a ∈ R. If ad(x)n = 0 for all x ∈ R, where n is a fixed positive integer, then ad(R) = 0 when R is an (n− 1)!-torsion free ring. In [18] Lee and Lin proved Bres̆ar’s result without the assumption of (n− 1)!-torsion free ring. Recently, Xu, Ma and Niu [20] Received October 9, 2007 and in revised form August 26, 2008. AMS Subject Classification: 16W20, 16W25, 16W55.