Non-free iterative differential modules

TL;DR

This paper explores non-free differential modules over differentially simple rings, providing examples and computing their Picard-Vessiot rings and Galois groups, extending the theory beyond free modules.

Contribution

It introduces explicit examples of non-free differential modules over differentially simple rings and computes their Picard-Vessiot rings and Galois groups, expanding the existing theory.

Findings

Examples of non-free differential modules are constructed.

Picard-Vessiot rings for these modules are explicitly computed.

Galois groups associated with these modules are determined.

Abstract

In the article "Picard-Vessiot theory of differentially simple rings" we established a Picard-Vessiot theory over differentially simple rings which may not be fields. Differential modules over such rings were proven to be locally free but do not have to be free as modules. In this article, we give a family of examples of non-free differential modules, and compute Picard-Vessiot rings as well as Galois groups for them.