Multilinear polynomials and cocentralizing conditions in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x ₁,..., x _n ) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x ₁, …, xn) on R. If F and G are generalized derivations of R such that F(x), x], G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds:

(1)	there exists α ∈ C such that F(x) = αx for all x ∈ R
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