Which integral domains $R$ with fraction field $K$ have the property that for any intermediate ring $R \subset S \subset K$, $S$ is Noetherian?
I'm not really good at naming things, but let's call such integral domains fractionally-closed Noetherian for short.
It's not difficult to see that any PID $R$ is fractionally-closed Noetherian, in fact, every subring of the fraction field of $R$ which contains $R$ is some localization of $R$. I have been able to prove at least a partial converse: if $R$ is a fractionally-closed Noetherian gcd-domain, then $R$ is a PID. Here is the proof:
Let $R$ be a gcd-domain which is not a PID. If $R$ is not Noetherian, $R$ is certainly not fractionally-closed Noetherian, so we are done. Suppose $R$ is Noetherian. We may find $a',b' \in R$ such that $(a',b')$ is not a principal ideal. Write $d = \mathrm{gcd}(a',b'), a' = ad, b' = bd$, then we have $\mathrm{gcd}(a,b)=1$ and $(d)(a,b)=(a',b')$ as ideals. If $(a,b)$ was principal, $(a',b')$ would be principal as well, so $(a,b)$ is not principal. Consider the ring
$S:=bR_a+R = \{b\frac{r_1}{a^n}+r_2|r_1,r_2 \in R, n \in \mathbb N_0\}$.
In $S$, we have the increasing chain of ideals $(\frac{b}{a}) \subset (\frac{b}{a^2}) \subset \dots$ To see that this chain is in fact properly increasing, it suffices to show that $\frac{1}{a} \notin S$ Suppose $\frac{1}{a}=b\frac{r_1}{a^n}+r_2$, where we may choose $r_1$ such that $\mathrm{gcd}(r_1,a)=1$. We must have $n \geq 1$, for else RHS is in $R$, but clearly $a$ is not a unit in $R$. Now $a^{n-1}-a^nr_2=br_1$. If $n=1$, this gives an easy contradiction as $(a,b) \neq (1)$, if $n \geq 2$, we get $a | br_1$, but we have $\mathrm{gcd}(a,br_1) = 1$ from $\mathrm{gcd}(a,b) = 1$ and $\mathrm{gcd}(a,r_1)=1$.
So far, I have not found an example of fractionally-closed Noetherian integral domain which is not a PID, so one might as well suspect that any such domain is in fact a PID, but I have not been able to prove this either.
My questions is: Can you prove or disprove that any fractionally-closed Noetherian integral domain is a PID or give some interesting necessary or sufficient criteria?