  We show that there exists a Galois correspondence between subalgebras of an
H-comodule algebra A over a base ring R and generalised quotients of a Hopf
algebra H if both A and H are flat Mittag--Leffler modules. We also provide new
criteria for subalgebras and generalised quotients to be closed elements of the
Galois connection constructed. We generalise the Schauenburg theory of
admissible objects to this more general context. Then we consider coextensions
of H-module coalgebras. We construct a Galois connection for them and we prove
that H-Galois coextensions are closed. We apply the results obtained to the
Hopf algebra itself and we give a simple proof that there is a bijective
correspondence between right ideal coideals of H and its left coideal
subalgebras when H is finite dimensional. Furthermore, we formulate a necessary
and sufficient condition for a bijective Galois correspondence for A=H. We also
consider crossed product algebras.
