  Integrals in Hopf algebras are an essential tool in studying finite
dimensional Hopf algebras and their action on algebras. Over fields it has been
shown by Sweedler that the existence of integrals in a Hopf algebra is
equivalent to the Hopf algebra being finite dimensional. In this paper we
examine how much of this is true for Hopf algebras over rings. We show that
over any commutative ring R that is not a field there exists a Hopf algebra H
over R containing a non-zero integral but not being finitely generated as
R-module. On the contrary we show that Sweedler's equivalence is still valid
for free Hopf algebras or projective Hopf algebras over integral domains.
Analogously for a left H-module algebra A we study the influence of non-zero
left A#H-linear maps from A to A#H on H being finitely generated as R-module.
Examples and application to separability are given.
