On selfinjectivity of Hopf algebras Any group algebra $kG$ of a finite group is selfinjective. More generally Gentile
 proves that for a group ring $RG$ with $R$ commutative and torsion free as a $\Bbb Z$-module, $RG$ is selfinjective if and only if $G$ is finite and $R$ is selfinjective. 
Any group ring is a Hopf algebra, and a finite-dimensional cocommutative Hopf algebra over a field is selfinjective. Is there a generalization of this result in the same spirit as the result of Gentile? (Or in greater generality, see the comments.)

Finite dimensional algebras have integrals, and using one you can show selfinjectivity at once —in fact, they are Frobenius. 
See for example the Lectures on Hopf algebras by Schneider, which Google will find for you, or Susan Montgomery's book in Hopf algebras.
