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

Robert Gilmer

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