Prime and Semiprime Rings with n-Commuting Generalized Skew Derivations

Let R be a ring, Q(r) the right Martindale quotient ring of R, C the extended centroid of R, L a noncentral Lie ideal of R, F a nonzero generalized skew derivation of R, and m, n, k >= 1 be fixed integers. We prove the following results: (a) Assume R is a prime. If either char(R) = 0 or char(R) > m + 1 and [F(u(m)), u(n)](k) = 0 for all u is an element of L, then there exists lambda is an element of C such that F(x) = lambda x, for all x is an element of R, unless when R satisfies s(4), the standard identity of degree 4. (b) Assume R is semiprime. If char(R) not equal 2 and [F(x), x(n)](k) = 0 for all x is an element of R, then either there exists lambda is an element of C such that F(x) = lambda x, for all x. R, or R contains a non-zero central ideal.