Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f(x 1,…, x n ) a noncentral multilinear polynomial over C in n noncommuting variables, and a, b ∈ R such that a[F(f(r 1,…, r n )), f(r 1,…, r n )]b = 0 for any r 1,…, r n ∈ R. Then one of the following holds: (1) a = 0; (2) b = 0; (3) there exists λ ∈ C such that F(x) = λx, for all x ∈ R; (4) there exist q ∈ U and λ ∈ C such that F(x) = (q + λ)x + xq, for all x ∈ R, and f(x 1,…, x n )(2) is central valued on R; (5) there exist q ∈ U and λ, μ ∈ C such that F(x) = (q + λ)x + xq, for all x ∈ R, and aq = μa, qb = μb.