Picard–Vessiot theory of differentially simple rings

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 generally 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.