Almost discrete valuation domains

TL;DR

This paper investigates a class of integral domains called almost valuation domains, exploring their properties, generalizations of DVRs, and characterizing the structure of complete local AV-domains and API-domains.

Contribution

It introduces and analyzes various generalizations of DVRs within almost valuation domains, including Noetherian, DVR-closure, and API-domains, with structural classifications.

Findings

Complete local AV-domains are characterized structurally.

AV-domains with DVR integral closure are identified.

Various generalizations of DVRs are systematically explored.

Abstract

Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b\in D\setminus \{0\}$ there exists a natural number $n$ with $a^{n}\mid b^{n}$ or $b^{n}\mid a^{n}$. AV-domains are closely related to valuation domains, for example, $D$ is an AV-domain if and only if the integral closure $\bar{D}$ is a valuation domain and $D\subseteq \bar{D}$ is a root extension. In this note we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with $\bar{D}$ a DVR, and quasilocal and local API-domains (i.e., for $\{a_{\alpha}\}_{\alpha\in \Lambda}\subseteq D$, there exists an $n$ with $(\{a_{\alpha}^{n}\}_{\alpha\in \Lambda})$ principal). The structure of complete local AV-domains and API-domains is determined.