N-centralizing generalized derivations on left ideals *

Let R be a prime ring with center Z(R), right Utumi quotient ring U and extended centroid C, S be a non-empty subset of R and n ≥ 1 a fixed integer. A mapping f : R → R is said to be n-centralizing on S if [f(x), xn] ε Z(R), for all x ε S. In this paper we will prove that if F is a non-zero generalized derivation of R, I a non-zero left ideal of R, n ≥ 1 a fixed integer such that F is n-centralizing on the set [I; I], then there exists α ε U and α ε C such that F(x) = xa, for all x ε R and I(α - α) = (0), unless when x1s4(x2, x3, x4, x5) is an identity for I.