Prüfer \(v\)-multiplication domains and related domains of the form \(D+D_S[\varGamma ^{*}]\)
type: Item
instance of: scholarly article
title: Prüfer \(v\)-multiplication domains and related domains of the form \(D+D_S[\varGamma ^{*}]\)
zbMATH Open document ID: 1192.13002
author: Gyu Whan Chang
author: Jung Wook Lim
author: B. G. Kang
published in: Journal of Algebra
publication date: 2010-06-08T00:00:00Z
review text: Let \(D\) be an integral domain, \(S\) be a saturated multiplicative subset of \(D\) with \(D\subsetneq D_S\) and \(\Gamma\) be a nonzero torsion-free (additive) grading monoid with \(\Gamma\cap-\Gamma=\{0\}\). Let \(D_S[\Gamma]\) be the semigroup ring of \(\Gamma\) over \(D_S\), \(\Gamma^{*}=\Gamma-\{0\}\), and \(D^{(S,\Gamma)}=D+D_S[\Gamma^{*}]=\{f\in D_S[\Gamma]|f(0)\in D\}\). When \(\Gamma=\mathbb{N}_0\), the additive monoid of non-negative integers, then \(D^{(S,\mathbb{N}_0)}=D+XD_S[X]\), where \(X\) is an indeterminate, has received a considerable interest by several authors. In particular it is known that when \(D^{(S,\mathbb{N}_0)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain). In the paper under review the authors determine when \(D^{(S,\Gamma)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain) in this more general setting. In fact they prove that \(D^{(S,\Gamma)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain) if and only if \(D\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain), \(\Gamma\) is a valuation semigroup and \(S\) is a \(t\)-splitting (resp., splitting, \(d\)-splitting) set of \(D\).
reviewed by: Parviz Sahandi
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 5718201
zbMATH Keywords: torsion-free grading monoid \(\Gamma\) with \(\Gamma\cap-\Gamma=\{0\}\)
zbMATH Keywords: \(D+D_S[\Gamma^*]\)
zbMATH Keywords: P\(v\)MD
zbMATH Keywords: \(t\)-splitting set
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.03.010
OpenAlex ID: W2313298070
cites work: The class group and local class group of an integral domain
cites work: Divisorial ideals and invertible ideals in a graded integral domain
cites work: Splitting the t-class group
cites work: The ring \(D+XD_S[X]\) and \(t\)-splitting sets
cites work: Homogeneous splitting sets of a graded integral domain
cites work: The class group of \(D+M\)
cites work: Q3987489
cites work: Almost splitting sets in integral domains
cites work: \(t\)-splitting sets in integral domains.
cites work: The construction \(D+XD_s[X]\)
cites work: On the class group of a graded domain
cites work: On the class group and the local class group of a pullback
cites work: Q3681045
cites work: Q3137901
cites work: Some Results on v-Multiplication Rings
cites work: Divisibility properties in semigroup rings
cites work: Q4255424
cites work: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
cites work: A remark on the unique factorization theorem
cites work: On finite conductor domains
cites work: The \(D+XD_ s[X]\) construction from GCD-domains
cites work: Q3436776
DOI: 10.1016/J.JALGEBRA.2010.03.010
description: scientific article; zbMATH DE number 5718201