Intersections of quasi-local domains

Prekowitz, Bruce

doi:10.1090/S0002-9947-1973-0319986-7

Intersections of quasi-local domains
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by Bruce Prekowitz PDF

Trans. Amer. Math. Soc. 181 (1973), 329-339 Request permission

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 $R = {V_1} \cap {V_2}$.

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