Generalized skew-derivations annihilating and centralizing on multilinear polynomials in prime rings

Let R be a prime ring of characteristic \(\ne 2\), \(Q_r\) its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) a generalized skew derivation of R, \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C not central-valued on R and S the set of all evaluations of \(f(x_1,\ldots ,x_n)\) in R. If \(a[F(x),x]\in C\) for all \(x\in S\), then there exist \(\lambda \in C\) and \(b\in Q_r\) such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\) and one of the following holds: (1) \(b\in C\); (2) \(f(x_1,\ldots ,x_n)^2 \) is central-valued on R; (3) R satisfies \(s_4\), the standered identity of degree 4.