Intersections of Quasi-Local Domains

Abstract

Let $R = \bigcap {{V_i}}$ be an intersection of quasi-local domains with a common quotient field $K$. Our goal is to find conditions on the ${V_i}$’s in order to get some or all of ${V_i}$’s to be localizations of $R$. We show for example that if ${V_1}$ is a $1$-dimensional valuation domain and if ${V_1} \nsupseteq {V_2}$, then both ${V_1}$ and ${V_2}$ are localizations of $R = {V_1} \cap {V_2}$.