Introduction to the Galois theory of Artinian simple module algebras
We give an introduction to a Galois theory of Artinian simple module algebras. To this end, we first recall the Picard-Vessiot theories of differential and difference equations, Umemura's differential Galois theory and Morikawa-Umemura's difference Galois theory. Then we sketch the main ideas of Amano and Masuoka's unification of the Picard-Vessiot theories of differential and difference extensions. We show how the differential Galois theory of Umemura and the difference Galois theory of Morikawa-Umemura can be unified using Artinian simple module algebras in lieu of differential or difference fields, respectively, and remove the restriction to fields of characteristic 0. Finally, we compare this unified theory to the Picard-Vessiot theory of Amano and Masuoka in the case of Picard-Vessiot extensions of Artinian simple module algebras.