Almost discrete valuation domains

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.