On the Set of $t$-Linked Overrings of an integral domain

Abstract

et $R$ be an integral domain with quotient field $L$. An overring $T$ of $R$ is $t$-linked over $R$ if $I^{-1}=R$ implies that $(T:IT)=T$ for each finitely generated ideal $I$ of $R$. Let $O_{t}(R)$ denotes the set of all $t$-linked overrings of $R$ and $O(R)$ the set of all overrings of $R$. The purpose of this paper is to study some finiteness conditions on the set $O_{t}(R)$. Particularly, we prove that if $O_{t}(R)$ is finite, then so is $O(R)$ and $O_{t}(R)=O(R)$, and if each chain of $t$-linked overrings of $R$ is finite, then each chain of overrings of $R$ is finite. This yields that the $t$-linked approach is more efficient than the Gilmer's treatment in \cite{G1}. We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain etc. We establish a connection between $O_{t}(R)$ and the set of all strongly divisorial ideals of $R$ and we conclude by a characterization of domains $R$ that are $t$-linked under all their overrings.