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 [greater than or equal to] 1 a fixed integer. A mapping f: R [right arrow] R is said to be n-centralizing on S if [f (x), [x.sup.n]] [member of] Z(R), for all x [member of] 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 [greater than or equal to] 1 a fixed integer such that F is n-centralizing on the set [I, I], then there exists a [member of] U and [delta] [member of] C such that F (x) = xa, for all x [member of] R and I(a - [delta]) = (0), unless when [x.sub.1][s.sub.4]([x.sub.2], [x.sub.3], [x.sub.4], [x.sub.5]) is an identity for I. Keywords and Phrases: Prime ring, Generalized derivation.
Throughout the paper unless specifically stated, R always denotes a prime ring with center Z(R) and extended centroid C, right Utumi quotient ring U. For any pair of elements x, [...]