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.