ANNIHILATORS OF POWER VALUES OF GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x(1), . . . . , x(n)) be a noncentral multilinear polynomial over C, in > I a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag (f (r(1), . . . , r(n)))(m) = 0 for all r(1), . . . , r(n) is an element of I, then one of the following holds: (1) aI = ag(I) = (0); (2) g(x) = qx, for some q is an element of U and aq I = 0; (3) [f(x(1), . . . , x(n)), x(n+1)]x(n+2) is an identity for I; (4) g(x) = cx + [q, x] for all x is an element of R, where c, q is an element of U such that cI = 0 and [q, I]I = 0.