TITLE: An example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime
QUESTION [8 upvotes]: I am looking for an example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime. 

In a Prüfer domain such an intersection is always a prime ideal.

REPLY [3 votes]: Let $(R,m)$ be a (commutative Noetherian) local ring which is not a domain. By Krull's intersection theorem, $\bigcap_{n=1}^{\infty} m^n=0$ is not a prime.   
One can use appropriate quotient of local domain to have local ring which is not a domain: $R=K[[X]]/(X^t)$