Radical perfectness of prime ideals in certain integral domains
type: Item
instance of: scholarly article
title: Radical perfectness of prime ideals in certain integral domains
zbMATH Open document ID: 1364.13005
DOI: 10.1216/JCA-2017-9-1-31
author: Gyu Whan Chang
author: Hwankoo Kim
published in: Journal of Commutative Algebra
publication date: 2017-04-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1491379318
review text: Let \(D\) be an integral domain. An ideal \(I\) of \(D\) is called radically perfect if the height of \(I\) is equal to the infimum of the number of generators of ideals of \(D\) whose radical is equal to the radical of \(I\). This generalizes of the notion of a set-theoretic complete intersection to non-Noetherian rings. In addition, \(D\) is called \(t\)-compactly packed if and only if every prime \(t\)-ideal of \(D\) is the radical of a principal ideal. Let \(T(D)\) be the set of \(t\)-invertible fractional \(t\)-ideals of \(D\). Then \(T(D)\) is an abelian group under the \(t\)-multiplication \(I*J = (IJ)t\). Hence, if we let \(P(D)\) be the subgroup of nonzero principal fractional ideals of \(T(D)\), then \(Cl(D) := T(D)/P(D)\), called the (\(t\)-)class group of \(D\), is an abelian group. The following theorem is one of the main result in the present paper: Theorem. If \(D\) is a UMT-domain, then \(D[X]\) is \(t\)-compactly packed if and only if \(D\) is a \(t\)-compactly packed domain with \(Cl(D[X])\) torsion. In addition, the authors introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 6702372
zbMATH Keywords: Prüfer domain
zbMATH Keywords: radically perfect
zbMATH Keywords: \(t\)-compactly packed
zbMATH Keywords: quasi-Prüfer domain
zbMATH Keywords: UMT-domain
zbMATH Keywords: SM domain
zbMATH Keywords: Serre's condition
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6702372