Is a ring whose every descending chain of prime ideals stabilizes, necessary Noetherian?

Question

It is know that the DCC on prime ideals holds in Noetherian rings (see e.g., This question). I ask whether the converse holds: Is a commutative ring whose every descending chain of prime ideals stabilizes Noetherian?

Answer 1

This is not true. Here you can see an example of a non Noetherian ring which has exactly one prime ideal, and so your condition trivially holds there.

All prime ideals of $R/I$ where $R$ is the infinite polynomial ring

Answer 2

The ring of algebraic integers is an example of a non-Noetherian ring with Krull dimension $1$, and so it would not have long chains of prime ideals.

Another example is an infinite direct product of fields: such a ring is von Neumann regular, and so it has Krull dimension $0$, but if it has infinitely many factors then it is not Noetherian.