Dp-minimal integral domains

TL;DR

This paper characterizes dp-minimal integral domains, showing they are local rings with specific valuation properties, and provides conditions under which such domains are valuation rings based on their residue fields.

Contribution

It establishes structural properties of dp-minimal integral domains and characterizes when they are valuation rings based on residue field conditions.

Findings

Dp-minimal integral domains are local rings.

Localizations at non-maximal primes are valuation rings.

A dp-minimal integral domain is a valuation ring if its residue field is infinite or finite with a principal maximal ideal.

Abstract

It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.