GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT IDEALS OF PRIME RINGS

WOS: 000334818000007Let R be a prime ring, f(x(1),...,x(n)) 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(r(1),...,r(n)))f(r(1),...,r(n)) is an element of C, for all r(1),...,r(n) is an element of I. If f(x(1),...,x(n)) is not central valued on R, then either char(R) = 2 and R satisfies s(4) or one of the following holds: (i) f (x(1),...,x(n))x(n+1) is an identity for I; (ii) F(I)I = (0); (iii) [f(x(1),...,x(n)),x(n+1)]x(n+2) is an identity for I, there exist b,c,q is an element of Q with q an invertible element such that F(x) = bx - qxq(-1) c for all x is an element of R, and q(-1)cI subset of I. Moreover, in this case either (b - c)I = (0) or b c is an element of C and f(x(1),...,x(n))(2) is central valued on R