I am looking for an example of the following situation:
- $R$ is a noetherian one-dimensional ring.
- $x \in R$ is a non-nilpotent element contained in infinitely many primes of $R$.
Obviously, nilpotent elements behave in such a way and I was asking myself if other elements can behave in a similar fashion.
If there is a reference that no such elements exist, please tell me or post a proof if you like to.
Thanks a lot in advance!