Abstract. Let H be a Hopf algebra and A an H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H, A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for H-simple H-module algebras.

The main result states that, under certain assumptions about a Hopf algebra H , every H -semiprime right Noetherian H -module algebra has a quasi-Frobenius classical right quotient ring. Another question treated in the paper is concerned with the extension of H -module structures to quotient rings. These results have an application to the semiprimeness problem for smash product algebras A # H .

Let $ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on $ and a scalar $, we introduce an associative algebra $ of dimension pn over K. The algebra $ is isomorphic to the reduced enveloping algebra $, while the algebra $ is nothing but the reduced symmetric algebra $. Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple $-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple $-modules for all $ having a toral stabiliser of a linear function.

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