N-Centralizing Generalized Derivations on Left Ideals

10.6988/TOJIMS.201211.0425

Asma Ali；Faiza Shujat；Vincenzo De Filippis

Prime ring ； Generalized derivation

Tamsui Oxford Journal of Information and Mathematical Sciences

28卷4期（2012 / 11 / 01）

425 - 436

英文

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); x^n] ∈Z(R), for all x ∈ 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 a∈U and α∈C such that F(x) = xa, for all x∈R and I(α-α) = (0), unless when x1s4(x2, x3, x4, x5) is an identity for I.