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≠a∈ R such that a(F(x)mF(y)n-ynxms = 0 ∀ x,y ⊖ L then R ⊆ M2(K), the 2 × 2 matrix ring over a field K. (b) If R is semiprime and (F(x)mF(y)n-ynxms = 0≠a∈x,y L then either L centralizes a nonzero ideal of R or [s4(x1,..,x4),x5] is a polynomial identity for R.