On actions of Hopf algebras with commutative coradical

Let H be an m-dimensional Hopf algebra with left integral t, let R be a lef t H-module algebra with 1 containing an element gamma with t --> gamma = 1, and let S = R-H. It is proved that R is fully integral over S, every simpl e right R-module has a length less than or equal to m over S and J(S)(m) su bset of or equal to J(R) boolean AND S subset of or equal to J(S), where J( R) is the Jacobson radical of R, provided that H is pointed. Finally, it is shown that if S is a PI algebra, then R is a PI algebra as well, provided that H has a cocommutative coradical. (C) 2001 Elsevier Science B.V. All ri ghts reserved.