Dp-finite and Noetherian NIP integral domains

TL;DR

This paper investigates NIP integral domains, especially Noetherian and finite dp-rank cases, establishing structural properties such as being henselian local rings and classifying dp-minimal Noetherian domains.

Contribution

It provides new structural results for NIP Noetherian domains and finite dp-rank integral domains, including classification and conditions for henselianity.

Findings

NIP Noetherian domains that are not fields are semilocal of Krull dimension 1

Fraction fields of such domains have characteristic 0

Integral domains of finite dp-rank are henselian local rings

Abstract

We prove some results on NIP integral domains, especially those that are Noetherian or have finite dp-rank. If $R$ is an NIP Noetherian domain that is not a field, then $R$ is a semilocal ring of Krull dimension 1, and the fraction field of $R$ has characteristic 0. Assuming the henselianity conjecture (on NIP valued fields), $R$ is a henselian local ring. Additionally, we show that integral domains of finite dp-rank are henselian local rings. Finally, we lay some groundwork for the study of Noetherian domains of finite dp-rank, and we classify dp-minimal Noetherian domains.