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
