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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 .

Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials