A domain $R$ is a Dedekind domain iff it is integrally closed, Noetherian with Krull dimension at most one. I think they put Noetherian in the definition not only to make proofs easier but also because it's important. To see that it's really necessary to have Noetherian in the definition do you have an example for me of a domain $R$ which is integrally closed, not Noetherian with Krull dimension one? All my examples of domains which are not Noetherian fail to have Krull dimension one.
2026-03-26 00:58:19.1774486699
Almost Dedekind (not Noetherian)
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