When the Zariski space is a Noetherian space

TL;DR

This paper characterizes conditions under which the Zariski space of valuation rings over a domain and its minimal elements are Noetherian, providing insights into their topological structure.

Contribution

It offers a characterization of when the Zariski space and its minimal elements are Noetherian, advancing understanding of their topological properties.

Findings

Zariski space is Noetherian under specific algebraic conditions

Minimal Zariski space elements are Noetherian in certain cases

Provides criteria linking algebraic properties to topological Noetherianity

Abstract

We characterize when the Zariski space $\mathrm{Zar}(K|D)$ (where $D$ is an integral domain, $K$ is a field containing $D$ and $D$ is integrally closed in $K$) and the set $\mathrm{Zar_{min}}(L|D)$ of its minimal elements are Noetherian spaces.