It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a valuation ring if it is integrally closed?
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1 Answer
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This is (essentially) a conjecture of Krull; a counter-example was given by P. Ribenboim: Sur une note de Nagata relative à un problème de Krull, Math. Zeit. 64, 159-168 (1956). Note that "primaire" means "local of dimension 1", and that "complètement intégralement clos" is stronger than "integrally closed".
