Some finiteness conditions on the set of overrings of an integral domain

Let $D$ be an integral domain with quotient field $K$ and integral closure $\overline D$. An overring of $D$ is a subring of $K$ containing $D$, and $\mathcal{O}(D)$ denotes the set of overrings of $D$. We consider primarily two finiteness conditions on $\mathcal{O}(D)$: (FO), which states that $\mathcal{O}(D)$ is finite, and (FC), the condition that each chain of distinct elements of $\mathcal{O}(D)$ is finite. (FO) is strictly stronger than (FC), but if $D=\overline{D}$, each of (FO) and (FC) is equivalent to the condition that $D$ is a Prüfer domain with finite prime spectrum. In general $D$ satisfies (FC) iff $\overline{D}$satisfies (FC) and all chains of subrings of $\overline{D}$ containing $D$have finite length. The corresponding statement for (FO) is also valid.