Irreducible actions and faithful actions of hopf algebras

Let H be a Hopf algebra acting on an algebra A. We will examine the relationship between A, the ring of invariants A ^H, and the smash product A # H. We begin by studying the situation where A is an irreducible A # H module and, as an application of our first main theorem, show that if D is a division ring then [D : D ^H]≦dim H. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness of A # H is strongly related to the faithfulness of the left and right actions of A # H on A.