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 _xed integer. A mapping f : R → R is said to be n-centralizing on S if [f(x); xn] 2 Z(R), for all x 2 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 _xed integer such that F is n-centralizing on the set [I; I], then there exists a ⊂ U and α⊂ C such that F(x) = xa, for all x ⊂ R and I(a-a) = (0), unless when x1s4(x2; x3; x4; x5) is an identity for I.