Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f (xj., ...,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f (r))f (r) - f (r)H(f (r)) e C, for all r = (r1,... ,rn) e In. 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 e R, and (a - b)I = (0) = (a + p - a)I; 2. R satisfies s4, the standard identity of degree 4, and there exist a, a1 e U, a, /3 e C such that G(x) = ax + xa1 + ax and H(x) = a'x - xa + /3x, for all x e R; 3. R satisfies s4 and there exist a, a1 e U, and d : R → R, a derivation of R, such that G(x) = ax + d(x) and H(x) = xa1 - d(x), for all x e R, with a + a1 e C; 4. R satisfies s4 and there exist a, a1 e U, and d : R - R, a derivation of R, such that G(x) = xa + d(x) and H(x) = ax1 - d(x), for all x e R, with a - a1 e C; 5. there exists e2 = e e S'oc(RC) such that I = eR and one of the following holds: (a) [f (x1,. .. , xn), xn+1]xn+2 is an identity for I; (b) char(R) = 2 and s4(x1,x2,x3,x4)x5 is an identity for I; (c) [f (x1,. .. , xn)2, xn+1 ]xn+2 is an identity for I and there exist a, a1, b, b1 e U, a e C and d : R - R, a derivation of R, such that G(x) = ax + xa1 + d(x), H(x) = bx + xb1 - d(x), for all x e R, with (a - b1 - α)I = (0) = (b - a1 - α)I. © Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology.