  This main purpose of this article is the unification of the Galois theory of
algebraic differential equations by Umemura and the Galois theory of algebraic
difference equations by Morikawa-Umemura in a common framework using Artinian
simple D-module algebras, where D is a bialgebra. We construct the Galois hull
of an extension of Artinian simple D-module algebras and define its Galois
group, which consists of infinitesimal coordinate transformations fulfilling
certain partial differential equations and which we call Umemura functor. We
eliminate the restriction to characteristic 0 from the above mentioned theories
and remove the limitation to field extensions in the theory of
Morikawa-Umemura, allowing also direct products of fields, which is essential
in the theory of difference equations. In order to compare our theory with the
Picard-Vessiot theory of Artinian simple D-module algebras due to Amano and
Masuoka, we first slightly generalize the definition and some results about
them in order to encompass as well non-inversive difference rings. Finally, we
give equivalent characterizations for smooth Picard-Vessiot extensions,
describe their Galois hull and show that their Umemura functor becomes
isomorphic to the formal scheme associated to the classical Galois group scheme
after a finite \'etale base extension.
