Let H be a finite dimensional Hopf algebra over the field k and let A be an H-module algebra. Then the smash product A#H is a ring extension of A and the ring of H-invariants defined by is a subring of A. This paper studies the connection between these rings using the key fact that if H is semisimple, then it has a left integral, namely an element x with for all , satisfying (x)
Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if are A#H-modules, then W is an A-direct summand of V if and only if it is an A#H-direct summand.
Section 2 introduces a Morita context where and with [, ] nondegenerate. Then criteria are given for (, ) to be nondegenerate and the context is used to relate to A#H. The paper also proposes an interesting question. Namely if H is a finite dimensional semisimple Hopf algebra and if A is semiprime, must A#H also be semiprime? Some partial results are given.