Locally Dedekind Rings
There exist integral domains which are locally but not globally Dedekind: the localization of at each maximal ideal is a Dedekind ring (equivalently, a DVR) but itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called "proper almost Dedekind rings."
Read more about this topic: Dedekind Domain
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