## Abstract

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R ƒ (x1… xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(ƒ (r)) ƒ (r)- ƒ (r)H(ƒ (r)) ∈ C, for all r =(r_{1},….,r_{n}) ∈ I^{n}. Then one of the following holds:

1. there exist a; b; p ∈ U and α C such that G(x)= ax + [p, x] and H(x) = bx, for all x ∈ R, and (a-b)I=(0)=(a + p- α)I;

2. R satisfies s_{4}, the standard identity of degree 4, and there exist a; a' ∈ U, α,β ∈ C such that G(x) =ax + xa' + αx and H(x) = a'x - xa +βx, for all x ∈ R;

3. R satisfies s_{4} and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = ax + d(x) and H(x)= xa'- d(x), for all x ∈ R, with a + a' ∈ C;

4. R satisfies s_{4} and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = xa + d(x) and H(x) = ax' - d(x), for all x ∈ R, with a - a' ∈ C;

5. there exists e^{2}= e ∈ Soc(RC) such that I = eR and one of the following holds:

(a) [ƒ (x_{1} ,…., x_{n}); x_{n + 1}] x_{n+2} is an identity for I;

(b) char (R) = 2 and s_{4}(x_{1}; x_{2}; x_{3}; x_{4})x_{5} is an identity for I;

(c) [ƒ (x_{1} , …, x_{n})^{2}; x_{n+1}]x_{n+2} is an identity for I and there exist a, a', b, b' ∈ U,α ∈ C and d : R → R, a derivation of R, such that G(x) = ax + xa' + d(x), H(x)=bx + xb' - d(x), for all x ∈ R, with (a - b' - α) I=(0)=( b-a'-α )I

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