On the set of t-linked overrings of an integral domain

Abstract

Let 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 Ot(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 Ot(R). Particularly, we prove that if Ot(R) is finite, then so is O(R) and Ot(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 (Proc Am Math Soc 131:2337–2346, 2002). 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 Ot(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.