When the Zariski space is a Noetherian space

Abstract

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