An Engel condition with generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2 with right Martindale quotient ring Q and extended centroid C. Let further k = 1 be a fixed integer, f (x1,..., xn) a multilinear polynomial over C which is not central-valued on R. If F : R. R is a nonzero generalized skew derivation of R such that [ F(f (r1,..., rn)), f (r1,..., rn)] k = 0 for all r1,..., rn. R, then either there exists.. C such that F(x) =.x for all x. R, or one of the following holds: (a) char(R) = p > 0 and f (x1,..., xn) ps is central-valued on R for a suitable s = 0; (b) there exist a, b. Q with a -b. C such that F(x) = ax + xb for all x. R, and f (x1,..., xn) 2 is central-valued on R.