On the structure of prime and semiprime rings with generalized skew derivations

Let R be a ring of characteristic different from 2, m,n,s >= 1 fixed positive integers, L a noncentral Lie ideal of R and F : R -> R a nonzero generalized skew derivation of R. We prove the following results: (a) If R is prime and there exists 0 not equal = a is an element of R such that a(F(x)F-m(y)(n)-y(n)x(m))(s)=0 for all x, y is an element of L then R subset of M (2)(K), the 2x2 matrix ring over a field K. (b) If R is semiprime and (F(x)F-m(y)(n)-y(n)x(m))s= 0 for all x, y is an element of L then either L centralizes a nonzero ideal of R or [s(4)(x(1),...,x(4)),x(5)]is a polynomial identity for R.