An Annihilator Condition With Generalized Derivations On Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x(1),..., x(n)) a noncentral multilinear polynomial over C. If there exists aR such that, for all r(1),..., r(n)R, a[F-2(f(r(1),..., r(n))), f(r(1),..., r(n))]=0, then one of the following statements hold: a=0; There exists C such that F(x)=x, for all xR; There exists cU such that F(x)=cx, for all xR, with c(2)C; There exists cU such that F(x)=xc, for all xR, with c(2)C.