GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMI ALS IN RIGHTIDEALS OF PRIME RINGS

Let R be a prime ring, f(x1, ..., xn) a multilinear polynomial over C in n noncommuting indeterminates, I a nonzero right ideal of R, and F : R → R be a nonzero generalized skew derivation of R. Suppose that F(f(r1, ..., rn))f(r1, ..., rn) in C, for all r1, ..., rn in I. If f(x1, ..., xn) is not central valued on R, then either char(R) = 2 and R satises s4 or one of the following holds: (i) f(x1, ..., xn)xn+1 is an identity for I, (ii) F(I)I = (0), (iii) [f(x1, ..., xn), xn+1]xn+2 is an identity for I, there exist b, c, q in Q with q an invertible element such that F(x) = bx - qxq^-1c for all x in R, and q^-1cI subseteq I. Moreover, in this case either (b - c)I = (0) or b - c in C and f(x1, ..., xn)² is central valued on R.