Composition and orthogonality of derivations with multilinear polynomials in prime rings

Abstract

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and $$\delta $$ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ over C which is not central valued. Let $$p,q\in R$$ . We prove the followings.