Rather large subsets and vanishing generalized derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and let f(x1,...,xn) be a multilinear polynomial over C, not central valued on R. Suppose that F and G are non-zero generalized derivations of R and 0≠u0 is an element of R such that u0F(G(f(r1,...,rn))f(r1,...,rn)) = 0 for all r1,...,rnε R. Then one of the following holds: there exists a,c∈U, the right Utumi quotient ring of R, such that F(x) = ax and G(x) = cx, for all x∈R, with u0ac = 0; there exists a∈U, the right Utumi quotient ring of R, such that F(x) = ax, for all x∈R, with u0a = 0; there exists a,b,c∈U, the right Utumi quotient ring of R, such that F(x) = ax+xb and G(x) = cx, for all x∈R, with u0c = u0ac=0; f(x1,...,xn)2 is central valued on R and there exists a,b,c∈U, such that F(x) = ax+xb, G(x) = cx, for all x∈R, with u0(ac+cb) = 0; there exists a,c∈U and d:R→R a derivation of R such that F(x) = ax+d(x) and G(x) = cx, for all x∈R, with u0c = u0 (ac+ d(c)) = 0. Moreover, in this case, d is not an inner derivation of R. © 2017, Copyright © Taylor & Francis.