Flatness of Noetherian Hopf algebras over coideal subalgebras

It is proved in the paper that a Noetherian residually finite-dimensional Hopf algebraHis a flat module over any right Noetherian right coideal subalgebraA. In the case whenAis a Hopf subalgebra we get faithful flatness. These results are obtained by verifying the existence of classical quotient rings ofAandH. It is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.