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Let  R  be a non commutative prime ring of characteristic dierent from 2,  U  be the Utumi quotient ring of  R  with the extended centroid  C ,  f ( x 1, … ,  x n) a multilinear polynomial over  C  which is not central valued on  R ,  f ( R ) the set of all evaluations of the polynomial  f  ( x1 , … ,  x n). Suppose  F  and  G  are two nonzero generalized derivations on  R  and let  p ,  q , ∈  R  be such that  f ( R ) satisfies the dierential identity  In this paper we prove that, if  R  does not satisfy the standard identity  s4 , then either  R  satisfies (1) or  f ( x1 , … ,  x n) 2  is central valued on  R . Moreover, in both cases we describe all possible forms of generalized derivations  F  and  G .

Some generalized identities on prime rings and their application for the solution of annihilating and centralizing problems