  There are theorems in the literature which are referred as Maschke type
theorems, because of the attempt to generalize Maschke's classical theorem. Our
remark is that all these theorems are in fact providing sufficient conditions
for the separability of a certain forgetful functor. But, unlike classical
Maschke's theorem, where the sufficient condition is also necessary, none of
these theorems gives any necessary condition for the separability of the
forgeful functor in question. Our job is to find a necessary and sufficient
condition for the separability of the forgetful functor from the category of
Doi-Hopf modules forgetting the comodule structure. As in the classical case,
this functor is separable iff exists a certain "integral map" with a
normalizing property. As particular cases, improved versions of "Maschke type"
theorems are given. Other various applications are given. For example, keeping
in mind that the category of Yetter-Drinfel'd modules is also a category of
Doi-Hopf modules, we obtain the notion of quantum integral. For H a finite
dimensional Hopf algebra, the existence of such a quantum integral is a
necessary and sufficient condition for the Drinfel'd double to be a separable
extension of H.
