Taft algebras acting on associative algebras

We examine actions of the n2-dimensional Taft algebra H on associative algebras R. We first determine when various H-stable subrings of R contain nonzero invariants. Then we look at sufficient and necessary conditions for smash product R#H to be either semiprime or prime. This enables us to prove that if H acts on a field K such that the skew derivation is nonzero, then K#H must be a direct sum of n copies of the n×n matrices over the invariant subfield KH.