To show a localisation is a DVR

Question

Let $R$ be an integral domain and $P$ a non-zero prime ideal in $R[X]$ such that the contraction to $R$ is $(0)$. Show that $R[X]_P$ is a DVR.

I know some equivalent criteria. See it's already local, if I can show it's a PID I'm done.

Answer 1

Since $P \cap R = 0$, $W = R \setminus P$ are units in $R[X]_P$. In other words, $R[X]_P$ is a furether locaization of $K[X]$ at $P' = PK[X]$, where $K = W^{-1}R = \operatorname{Quot} R$ which is a field. In the PID $K[X]$, the localization at any non-zero prime ideal is a DVR.