Finite unions of overrings of an integral domain

Abstract

Let $R$ be an integral domain, and let $A,A_1,A_2,\dots ,A_s$ be overrings of $R$, where $A$ is of the form $S^{-1}R$, where $S=R\setminus {\mathfrak{𝔭}}_1\cup \cdots \cup {\mathfrak{𝔭}}_n$ for some prime ideals ${\mathfrak{𝔭}}_i$, and where each $A_i$, $i\ge 2$, is of the form $S_i^{-1}R$ for some multiplicatively closed subset $S_i$ of $R$. It is shown that if $A\subseteq A_1\cup \cdots \cup A_s$, then $A\subseteq A_i$ for some $i$.