I am currently learning about dedekind domains, and there seem to be several equivalent definitions. One definition asks that the ring should be Noetherian, integrally closed, and that its prime ideals should be maximal. The first and third conditions make enough sense to me (since the other definitions of dedekind domain are mainly about its ideals) but I do not understand the importance of the integrally closed condition. Is there an intuitive explanation for this? I can see where it appears in the proof that these definitions are equivalent, but I'm not getting much from that. Any help is appreciated.
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A counterexample to integral closure immediately yields a counterexample to unique factorizations of ideals, e.g. see here in the linked dupe. – Bill Dubuque Dec 07 '23 at 05:20
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It is hard to tell when I'm asking a duplicate question! I search the website beforehand... – Ray James Dec 07 '23 at 05:24