Commuting Values of Generalized Derivations On Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x(1),..., x(n)) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x(1),..., x(n)) in R. If [G(u)u, G(v)v]=0, for any u, vf(R), we prove that there exists cU such that G(x)=cx, for all xR and one of the following holds: f(x(1),..., x(n))(2) is central valued on R; R satisfies s(4), the standard identity of degree 4.