# Existance of uniformizing element in a localization of a Dedekind domain

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.