On finitely stable domains

TL;DR

This paper investigates properties of Archimedean and locally Archimedean stable domains, establishing conditions under which they are finitely stable, Mori, or satisfy ACCP, and providing counterexamples for certain local properties.

Contribution

It characterizes stable domains in relation to finiteness, Mori property, and Archimedean conditions, and presents new examples and counterexamples in the theory of stable domains.

Findings

A one-dimensional stable domain is finitely stable and Mori.

Locally Archimedean stable domains satisfy ACCP.

Archimedean stable semilocal domains are locally Archimedean.

Abstract

We study Archimedean and locally Archimedean stable domains. We prove that a domain is stable and one-dimensional if and only if it is finitely stable and Mori. But we give examples of Archimedean stable local domains that are not one-dimensional. We also prove that a locally Archimedean stable domain satisfies accp and that Archimedean stable semilocal domains are locally Archimedean. But generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.