Question
Let $R$ be an integral domain and $P$ a non zero prime ideal in $R[x]$ such that $P\cap R=(0)$. Then show that $R[x]_{P}$ is a discrete valuation ring.
Attempt
$R[x]_{P}$ is a local ring and an integral domain as $R$ is. However it is not clear to me how every ideal of $R[x]_{P}$ is principal or how is this ring even Noetherian.
Any help is appreciated.