On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t >= 1 fixed integers with t <= m + n + s. Suppose that alpha is a non-trivial automorphism of R and let Phi(x,y) = [x, y](t) - [x y](m) [alpha( [x ,y]), [x,y]](n) [x,y](s). Thus, (a) if Phi(u, v) = 0 for any u,v is an element of L, then L subset of Z(R); (b) if Phi(u,v) is an element of Z(R) for any u, v is an element of L, then either L subset of Z(R) or R satisfies s(4), the standard identity of degree 4. We also extend the results to semiprime rings.