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 μ is a nonzero skew derivation of R with associated automorphism α and that f(x1; ⋯ xn) is a multilinear polynomial over C with n non-commuting variables. If [μ(f(r1; ⋯ rn)); f(r1; ⋯ rn)] 2 Z(R) for all r1; ⋯ rn 2 I, then there exists an idempotent element e 2 Q such that RCe = IC and f(x1; ⋯ xn) is central valued on eRCe.