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, f(x1, … , x_n ) a multilinear polynomial over C which is not central valued on R, f(R) the set of all evaluations of the polynomial f (x ₁, … , x_n ). Suppose F and G are two nonzero generalized derivations on R and let p, q, ∈ R be such that f(R) satisfies the differential identity