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.Comment: 14 page

Topics: Mathematics - Commutative Algebra, 13G05, 13F05 (Primary), 13B02, 13B22 (Secondary)
Year: 2007
OAI identifier: oai:arXiv.org:math/0611556

Suggested articles

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.