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

Let R be prime ring of characteristic different from 2 and F, G, and H are generalized derivations on R. Let U be Utumi quotient ring of R with extended centroid C and be a multilinear polynomial over C which is not central valued on R. If for all and give the complete possible forms of the mappings F, G, and H. [ABSTRACT FROM AUTHOR]