# Localization of a Dedekind ring is a Dedekind ring or a field

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.

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