Prime ideals in Hopf galois extensions

For a finite-dimensional Hopf algebraH, we study the prime ideals in a faithfully flatH-Hopf-Galois extensionR ⊂A. One application is to quotients of Hopf algebras which arise in the theory of quantum groups at a root of 1. For the Krull relations betweenR andA, we obtain our best results whenH is semisolvable; these results generalize earlier known results for crossed products for a group action and for algebras graded by a finite group. We also show that ifH is semisimple and semisolvable, thenA is semiprime providedR isH-semiprime.