I have the following definition of a Dedekind ring $R$:

$R$ is noetherian, one-dimensional (Krull dimension) and normal (e.g. integrally closed and an integral domain)

I want to show that any localization of a Dedekind ring is again a Dedekind ring or a field. I know that every localization $R_S$ of a Dedekind ring is again noetherian and normal and $\dim R_S\leq \dim R = 1$. If $\dim R_S=1$ I'm done, but what about $\dim R_S=0$? Then $R_S$ is noetherian of dimension $0$, hence artinian. But is every artinian normal ring a field or is there a different approach to my question?

Any help will be greatly appreciated.

  • 2
    $\begingroup$ Any Artinian integral domain is a field. $\endgroup$ – Lord Shark the Unknown Dec 18 '17 at 17:36
  • $\begingroup$ Oh since $(0)$ is a prime ideal and so has to be maximal. That was stupid. Thank you! $\endgroup$ – Luke Mathwalker Dec 18 '17 at 20:52
  • $\begingroup$ So even every zero-dimensional integral domain is a field, right? $\endgroup$ – Luke Mathwalker Dec 18 '17 at 20:56
  • 1
    $\begingroup$ @LukeMathwalker yes, you're right. $\endgroup$ – Xam Dec 19 '17 at 18:46

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