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?



Is it a valuation ring?
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*](https://link.springer.com/article/10.1007/BF01166564), Math. Zeit. 64, 159-168 (1956). Note that "primaire" means "local of dimension 1", and that "complètement intégralement clos" is stronger than "integrally closed".


