# Non-Noetherian Domain Which is Locally Noetherian

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.

• No, but an example is not easily found. Look (with Google) for an example of almost Dedekind domain which is not Dedekind. – user26857 Apr 24 at 17:19
• 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. – rschwieb Apr 24 at 19:22
• 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. – rschwieb Apr 24 at 19:30
• @rschwieb If I remember well, Example 59 from Hutchins is such an example. – user26857 Apr 24 at 22:21
• @user26857 Thanks much for the reference! I will take a look. – rschwieb Apr 25 at 13:21