On actions of Hopf algebras with cocommutative coradical

Let H be an m-dimensional Hopf algebra with left integral t, let R be a left H-module algebra with 1 containing an element γ with trightwards wave arrowγ=1, and let S=RH. It is proved that R is fully integral over S, every simple right R-module has a length ≤m over S and J(S)msubset of or equal toJ(R)∩Ssubset of or equal toJ(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.