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I want to prove that the following definition of a Dedekind domain are equivalent:

$(i)$ $R$ is noetherian and integrally closed, and every nonzero prime ideal is maximal.

$(ii)$ $R$ is noetherian, and each localization $R_P$ is a discrete valuation ring.

I know that the valuation of $R_P$ is a $\pi$-adic valuation where $\pi$ is a uniformizer of $P$ element, but I want to prove the existance of that uniformizing element.

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