Identities with Generalized Derivations and Multilinear Polynomials

Abstract

Let R be a prime ring of characteristic different from 2, U be its Utumi quotient ring with extended centroid C and $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations on R such that $$F^2(f(r))f(r)-G(f(r)^2)=0$$ for all $$r=(r_1,\ldots ,r_n)\in R^{n}$$. Then one of the following holds: