Let $R$ be a Noetherian ring. If $t\in R$ generates a prime ideal in $R$ show that $R$ is normal iff $R_t$ is normal and $R_{(t)}$ is a DVR.
I'm trying to solve the converse of this exercise. Let $\mathfrak p$ be a prime ideal in $R$. If $t \not \in \mathfrak p$ then I have proved that $R_{\mathfrak p}$ is an integrally closed domain. If $t \in \mathfrak p$ and $\mathrm{ht}(\mathfrak p)=1$ then $R_{\mathfrak p}$ is an integrally closed domain. But I am having trouble in proving this in the case $t \in \mathfrak p$ and $\mathrm{ht}(\mathfrak p) \geq2$.
Any hints/ideas.