Picard-Vessiot theory of differentially simple rings

TL;DR

This paper extends Picard-Vessiot theory to include differentially simple rings as bases, broadening the classical Galois theory from fields to more general rings, and establishing a new framework for differential Galois theory.

Contribution

It develops a new Picard-Vessiot theory framework where the base is a differentially simple ring instead of a field, expanding the scope of differential Galois theory.

Findings

Established a Picard-Vessiot theory over differentially simple rings.

Demonstrated the existence of Picard-Vessiot rings in this broader setting.

Connected classical Galois theory with the new ring-based approach.

Abstract

In Picard-Vessiot theory, the Galois theory for linear differential equations, the Picard-Vessiot ring plays an important role, since it is the Picard-Vessiot ring which is a torsor (principal homogeneous space) for the Galois group (scheme). Like fields are simple rings having only (0) and (1) as ideals, the Picard-Vessiot ring is a differentially simple ring, i.e. a differential ring having only (0) and (1) as differential ideals. Having in mind that the classical Galois theory is a theory of extensions of fields, i.e. of simple rings, it is quite natural to ask whether one can also set up a Picard-Vessiot theory where the base is not a differential field, but more general a differentially simple ring. It is the aim of this article to give a positive answer to this question, i.e. to set up such a differential Galois theory.