Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

For a Hopf algebra H and an H-module algebra A module-finite over its center it is proved that there is an equivalence relation on a subset SpecfA of the prime spectrum of A which exactly corresponds to the orbit relation in case of group actions. A linearly compact topologically H-simple H-module algebra LP(A) have been associated with each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P\in\mathop{\rm Spec}_fA$\end{document}. When A is noetherian and H-semiprime, it is shown that A has a quasi-Frobenius classical quotient ring.