Conversely, the valuation \ nu : A \ rightarrow \ Z \ cup \ { \ infty \ } on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K = \ text { Quot } ( A ); the associated discrete valuation ring \ mathcal { O } _ K is just A.
32.
Then " A " is integrally closed if and only if ( i ) " A " is the intersection of all localizations A _ \ mathfrak { p } over prime ideals \ mathfrak { p } of height 1 and ( ii ) the localization A _ \ mathfrak { p } at a prime ideal \ mathfrak { p } of height 1 is a discrete valuation ring.
33.
Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring ( DVR ), so the same local characterization cannot hold for PIDs : rather, one may say that the concept of a Dedekind ring is the "'globalization "'of that of a DVR.
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