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This main purpose of this article is the unification of the Galois theory ofalgebraic differential equations by Umemura and the Galois theory of algebraicdifference equations by Morikawa-Umemura in a common framework using Artiniansimple D-module algebras, where D is a bialgebra. We construct the Galois hullof an extension of Artinian simple D-module algebras and define its Galoisgroup, which consists of infinitesimal coordinate transformations fulfillingcertain partial differential equations and which we call Umemura functor. Weeliminate the restriction to characteristic 0 from the above mentioned theoriesand remove the limitation to field extensions in the theory ofMorikawa-Umemura, allowing also direct products of fields, which is essentialin the theory of difference equations. In order to compare our theory with thePicard-Vessiot theory of Artinian simple D-module algebras due to Amano andMasuoka, we first slightly generalize the definition and some results aboutthem in order to encompass as well non-inversive difference rings. Finally, wegive equivalent characterizations for smooth Picard-Vessiot extensions,describe their Galois hull and show that their Umemura functor becomesisomorphic to the formal scheme associated to the classical Galois group schemeafter a finite \'etale base extension.

Galois theory of Artinian simple module algebras