  Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and
$A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules
and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can
also strengthen the Blattner-Montgomery duality theorem. Finally, we give
sufficient conditions to get the purity of $A^\circ$ in $R^A$.
