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Let $R$ be an integral domain such that the localization, $R_{\mathfrak p}$, at each prime ideal, $\mathfrak p \le R$ is Noetherian. Then is $R$ necessarily Noetherian?

In the case of $R$ not neccesarily being a domain, I know of a couple of examples to show that $R$ need not being Noetherian (e.g. this & this). However, in each example, $R$ is not a domain.

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    $\begingroup$ No, but an example is not easily found. Look (with Google) for an example of almost Dedekind domain which is not Dedekind. $\endgroup$ – user26857 Apr 24 at 17:19
  • $\begingroup$ Loper, K. Allan. "Almost Dedekind domains which are not Dedekind." Multiplicative ideal theory in commutative algebra. Springer, Boston, MA, 2006. 279-292 has a bunch, but as I ran down the list they all seem quite involved. Looks like it takes some work. $\endgroup$ – rschwieb Apr 24 at 19:22
  • $\begingroup$ Although I just got to the very end of the list, and it looks like perhaps the group ring of the group $(\mathbb Q,+)$ over the field $\mathbb Q$ might be a concisely described example. $\endgroup$ – rschwieb Apr 24 at 19:30
  • $\begingroup$ @rschwieb If I remember well, Example 59 from Hutchins is such an example. $\endgroup$ – user26857 Apr 24 at 22:21
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    $\begingroup$ @user26857 Thanks much for the reference! I will take a look. $\endgroup$ – rschwieb Apr 25 at 13:21

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