A note on generalized skew-derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, Q(r) its right Martindale quotient ring and C its extended centroid, f (X-1, . . . , X-n) a multilinear polynomial over C that is noncentral-valued on R and F a generalized skew derivation of R. If for some 0 not equal a is an element of R - C, a[F (f (r)), f (r)] - [F (f (r)), f (r)]a is an element of Z(R) for all r = (r(1), . . . , r(n)) is an element of R-n then one of the following conditions holds: (1) there exists lambda is an element of C such that F (x) = lambda x for all x is an element of R; (2) there exist b is an element of Qr and lambda is an element of C such that F (x) = bx + xb + lambda x for all x is an element of R and f (X-1, . . . , X-n) is central valued on R. As an application of this result, we investigate the commutator [S, T ] is an element of Z(R), where S = {[F (u), u]|u is an element of f (R)} and T = {[G(v), v]|v is an element of f (R)}, F and G two generalized skew derivations of R.