Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes?
It would be good if I had this result. As it would finish off my proof that the minimal primes of an ideal $I$ of a Noetherian ring $R$ consist of the minimal (w.r.t. inclusion) elements of $\text{Ass }I$.
Yes: if $\mathfrak p$ is a prime ideal generated by $n$ elements in a Noetherian ring, then it has height at most $n$. In other words, the longest strictly descending chain of prime ideals starting with $\mathfrak p$ has no more than $n$ steps. This follows from repeated application of Krull's principal ideal theorem.
That being said, there could be arbitrarily long finite descending chains of prime ideals in a Noetherian ring. Indeed, there are Noetherian rings of infinite Krull dimension.