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Can someone give me, with proof, an example of a Noetherian ring which has Krull dimension one but is not a Dedekind domain?

I think it would also be instructive to see other "near misses."

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    @vadim123 Only 15 topics tagged as Krull-dimension? I bet you can find more!2013-06-21

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The ring $\mathbb{Z}[2i]$ is an example. It satisfies all the properties of a Dedekind domain except that it is not integrally closed. (To see that it satisfies these properties, note that it is integral over $\mathbb{Z}$, so the Krull dimension is one, as integral extensions preserve dimension. It is also clearly noetherian.)

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    @Tony: That's the standard argument, I believe; I'm afraid I know of no others.2011-01-04
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The ring $R_1 = \mathbb{Z} \times \mathbb{Z}$ gives a counterexample to your claim (it is not a domain). However $\operatorname{Spec}(R_1)$ is a Dedekind scheme, so this is a somewhat cheap counterexample. The counterexample $R_2 = \mathbb{Z}[t]/(t^2)$ is more serious.

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If by "other near misses" you mean other rings that satisfy two of the three conditions for being a Dedekind domain, another such ring is k[x, y], which is Noetherian (by the Hilbert basis theorem), integrally closed (it is a UFD, and UFDs are integrally closed) but not every prime ideal is maximal (for example (x)).