Some generalized identities on prime rings and their application for the solution of annihilating and centralizing problems

Abstract

Let R be a non commutative prime ring of characteristic different from 2, U be the Utumi quotient ring of R with the extended centroid C, ^ƒ(x_1,..., x_n) a multilinear polynomial over C which is not central valued on R, ƒ(R) the set of all evaluations of the polynomial ƒ(x_1,...,x_n). Suppose F and G are two non-zero generalized derivations on R and let p, q, ∈  R be such that ƒ(R) satisfies the
differential identity 
(1)                     p[F(x); x] + q[G(x); x]p + [F(x), x]q.

In this paper we prove that, if R does not satisfy the standard identity _s4, then either R satisfies (1) or ƒ(x_1,..., x_n)² is central valued on R. Moreover, in both cases we describe all possible forms of generalized derivations F and G.