GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT

Abstract

Let R be a prime ring, f (x, . . . , x) a multilinear polynomial over Cin n noncommuting indeterminates, I a nonzero right ideal of R, andF : R-> R be a nonzero generalized skew derivation of R.Suppose that F (f (r, . . . , rn))f (r, . . . , r)? C, for all r1, . . . , rn? I.If f (x1, . . . , xn) is not central valued on R, then either char(R) = 2and R satisfies s4or one of the following holds:(i) f (x1, . . . , xn)xn+1is an identity for I;(ii) F (I)I = (0);(iii) [f (x1, . . . , xn), xn+1]xn+2is an identity for I, there existb, c, q? Q with q an invertible element such that F (x) =bx- qxq-1c for all x? R, and q-1cI? I. Moreover, inthis case either (b- c)I = (0) or b - c ? C and f(x1, . . . , xn)2is central valued on R.