Actions of divided power Hopf algebras

We study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power Hopf algebras act. These actions correspond to higher derivations. First, we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring, where the Lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic. We give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring. Then using an explicit construction of a duality theorem of Blattner and Montgomery in this context, we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring, under a locally nilpotent divided power Hopf algebra action.