This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions.
A ring $R$ is called universally catenary if every $R$-algebra of finite type is catenary. (Note that $R$ need not be noetherian.)
A ring $R$ is called a going-between ring if for every integral ring extension $R\subseteq S$, every saturated chain of primes in $S$ contracts to a saturated chain of primes in $R$.
From results by Ratliff we get the following.
Theorem: Every noetherian universally catenary ring is a going-between ring.
The proof is rather complicated, as it relies on several non-trivial results concerning relations between different chain conditions. In particular, it is not clear to me whether one can get through without noetherianness.
Is it known whether or not we can omit the noetherian hypothesis in the above result?