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.