Some results for generalized lie ideals in prime rings with derivation II

Let R be a prime ring of characteristic different from two, d : R ? R a non-zero derivation, and M a non-zero left ideal of R. We prove the following results: (1) if a ? R and [d(R), a]?, ? = 0, then ? (a) + ?(a) ? Z, the center of R, (2) if d([R, a]?, ?) = 0, then ? (a)+?(a) ? Z, (3) if ([R,M]?, ?, a)?, ? = 0, then a ? Z, (4) d(R), a) = 0 if, and only if, d((R, a)) = 0.