On \(S\)-strong Mori domains
type: Item
instance of: scholarly article
title: On \(S\)-strong Mori domains
zbMATH Open document ID: 1304.13038
author: Hwankoo Kim
author: Myeong Og Kim
author: Jung Wook Lim
published in: Journal of Algebra
publication date: 2014-09-04T00:00:00Z
reviewed by: Nikolay I. Kryuchkov
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 6339544
zbMATH Keywords: \(S\)-strong Mori domain
zbMATH Keywords: \(S\)-factorial domain
zbMATH Keywords: \(S\)-strong Mori module
zbMATH Keywords: Krull domain
Wikidata QID: Q112881575
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.06.015
OpenAlex ID: W1982908994
cites work: The ring \(D+XD_S[X]\) and \(t\)-splitting sets
cites work: Two star-operations and their induced lattices
cites work: S-NOETHERIAN RINGS
cites work: Anti-archimedean rings and power series rings
cites work: Agreeable domains
cites work: Q3137901
cites work: Some Results on v-Multiplication Rings
cites work: Some remarks on star-operations
cites work: A note on *_{𝑤}-Noetherian domains
cites work: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
cites work: On the converse of a well-known fact about Krull domains
cites work: Q5603302
cites work: Module-Theoretic Characterizations oft-Linkative Domains
cites work: Module-theoretic Characterizations of Strongly t-linked Extensions
cites work: \(S\)-Noetherian properties on amalgamated algebras along an ideal
cites work: Q3538810
cites work: Integral Domains in Which Each Ideal Is aW-Ideal
cites work: Power series rings over strong Mori domains.
cites work: Q4344301
cites work: w-DIMENSION OF DOMAINS. II
cites work: On w-modules over strong mori domains
cites work: On strong Mori domains
cites work: ω-MODULES OVER COMMUTATIVE RINGS
cites work: Well behaved prime t-ideals
cites work: Q4902017
DOI: 10.1016/J.JALGEBRA.2014.06.015
review text: Throughout this review \(D\) is an integral domain and \(S\) is a (not necessarily saturated) multiplicative subset of \(D\).NEWLINENEWLINE\textit{D. D. Anderson} and \textit{T. Dumitrescu} [Commun. Algebra 30, No. 9, 4407--4416 (2002; Zbl 1060.13007)] introduced the concepts of \(S\)-Noetherian rings and \(S\)-principal ideal rings. A commutative ring \(R\) is called \(S\)-Noetherian ring (\(S\)-principal ideal ring) if each ideal of \(R\) is \(S\)-finite (respectively \(S\)-principal), i.~e., for each ideal \(I\) of \(R\) , there exist an \(s\in S\) and a finitely generated (respectively principal) ideal \(J\) of \(R\) such that \(sI\subset J\subset I\).NEWLINENEWLINEThe main purpose of the paper under review is to introduce and to investigate the notions of \(S\)-strong Mori domains, \(S\)-strong Mori modules and \(S\)-factorial domains. An integral domain \(D\) is called an \(S\)-strong Mori domain (\(S\)-factorial domain) if each ideal of \(D\) is \(S\)-\(w\)-finite (respectively, \(S\)-\(w\)-principal). A fractional ideal \(I\) of \(D\) is called of \(w\)-finite type if \(I=J_w\) for some finitely generated ideal \(J\) of \(D\) (\(J_w=\{x\in K|xF\subseteqq J\}\) for some finitely generated ideal \(F\) of \(D\) such that \((F^{-1})^{-1}=D\) and \(K\) is the quotient field of \(D\)).NEWLINENEWLINEIn Section 1 the authors study basic properties of \(S\)-strong Mori domains and \(S\)-factorial domains.NEWLINENEWLINEIn Section 2 they prove that if \(S\) is an anti-Archimedean subset of \(D\), then \(D\) is an \(S\)-strong Mori if and only if the polynomial ring \(D[X]\) is an \(S\)-strong Mori domain, if and only if the \(t\)-Nagata ring \(D[X]_{N_v}\) is an \(S\)-strong Mori domain, if and only if \(D[X]_{N_v}\) is an \(S\)-Noetherian domain.NEWLINENEWLINEThe Section~3 is devoted to study the Cohen type theorem for \(S\)-factorial domains and to give the new characterizations of Krull domains.
description: scientific article; zbMATH DE number 6339544