Smash products, inner actions and quotient rings

Let if be a Hopf algebra over a field k, and A an H-module algebra over k. Let A^H = (a ∈ A|h · a = ε(h)a, all h ∈ H). This paper is mainly concerned with inner actions. We prove the existence of a "symmetric" quotient ring Q of A, which is also an H-module algebra, and consider Q-inner actions, an analogue of X-inner automorphisms. Under certain conditions on A and H we show that Q contains B, a finite-dimensional separable algebra over its center C, a field. Moreover, the centralizer of B in Q is Q^H. This is used to prove that if A^H is P.I. then so is A, and that A is fully integral over A^HC of bounded degree. We also consider connections between the A, A^H and A#H module structures.