Locally nilpotent skew derivations with central invariants

Let δ be a locally nilpotent q-skew derivation of an algebra R such that the invariants are central. With some natural assumptions on the q-characteristic, we show that if R is semiprime then R is commutative. We also examine other conditions which imply, even when R is not commutative, that the commutator ideal is contained in the prime radical. These results extend previous work of the authors and of Osterburg and may shed some light on a conjecture of Herstein.