On a notion of "Galois closure" for extensions of rings

TL;DR

This paper introduces a new concept of Galois closure for ring extensions, generalizing the classical notion from field theory, and explores its properties and behavior across different classes of rings.

Contribution

It defines a Galois closure for ring extensions, proves its functoriality and base change compatibility, and compares it with the classical case for fields.

Findings

Galois closure coincides with classical in field extensions

The construction is functorial and respects base change

Behavior analyzed for various ring classes

Abstract

We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.