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 .