Show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.
The only idea that I come to mind is, we know $PR_P$ is the maximal ideal of $R_P$. Since $P$ is a prime ideal of $R$ then $PR_P$ also is a prime ideal of $R_P$. hence $PR_P$ also is the only prime ideal of $R_P$. Since the radical ideal is the intersection of all the prime ideals, $PR_P$ also is a radical ideal of $R_P$.
I don't know if that would help for the proof, and I am not sure how to carry on. Please help. Thank you.