The classification of dp-minimal integral domains

TL;DR

This paper classifies dp-minimal integral domains, showing they are either fields, valuation rings, or constructed via a specific extension involving valuation overrings and finite subrings.

Contribution

It extends the classification of dp-minimal structures to integral domains, providing a detailed characterization and new construction methods.

Findings

Dp-minimal integral domains are either fields, valuation rings, or constructed from valuation overrings and finite subrings.

The classification builds on existing results for dp-minimal fields and valuation rings.

A new construction method for dp-minimal integral domains is introduced.

Abstract

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O extending R, a proper ideal I in O, and a finite subring S in O/I such that R is the preimage of S in O.