Posner's Second Theorem For Skew Derivations On Multilinear Polynomials On Left Ideals

Let R be a prime ring of characteristic different from 2 with symmetric Martindale quotient ring Q and extended centroid C and let I be a nonzero left ideal of R. Suppose that mu is a nonzero skew derivation of R with associated automorphism alpha and that f(x(1), . . . , x(n)) is a multilinear polynomial over C with n non-commuting variables. If [mu(f (r(1,) . . . , r(n))), f(r(1,) . . . , r(n))] is an element of Z(R) for all r(1), . . . , r(n) is an element of I, then there exists an idempotent element e is an element of Q such that RCe = IC and f(x(1), . . . , x(n)) is central valued on eRCe.