# Explanation on discrete valuation rings

Let $$A$$ be a noetherian domain. We want to prove that if $$A$$ is normal then, for every prime ideal $$p$$ associated to a principal ideal of $$A$$, the localization $$A_p$$ is a discrete valuation ring. It suffices to prove that, for every prime ideal $$p$$ associated to a principal ideal of $$A$$, the ring $$A_p$$ is a principal ideal domain.

I don't understand why the sentence in italics is true: the only proposition that I studied, that seems relevant, in this context would say that if $$pA_p$$ is principal and $$A_p$$ has dimension $$1$$ then $$A_p$$ is a DVR. The first two things that I thought are that either the localization at a prime ideal associated to a principal ideal has dimension $$1$$, or that a local Noetherian domain that is a principal ideal domain is also a DVR. However it is strange that the notes don't even mention if we are using one of these two arguments; does one of them (at least) actually hold? If not, how do you justify the sentence in italics? Thanks