Almost splitting sets in integral domains
type: Item
instance of: scholarly article
title: Almost splitting sets in integral domains
zbMATH Open document ID: 1091.13001
author: Gyu Whan Chang
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-18T00:00:00Z
review text: Let \(S\) be a saturated multiplicatively closed subset of an integral domain \(A\) and \(N(S)= \{0\neq x\in A\mid (x,s)_v= A\) for all \(s\in S\}\). Then \(S\) is called an almost splitting set if for each \(0\neq d\in A\), there is an integer \(n= n(d)- 1\) such that \(d^n= st\) for some \(s\in S\) and \(t\in N(S)\). In this paper, the author studies almost splitting sets and domains of the form \(A+ X^2B[X]\), where \(B\) is an overring of \(A\). He shows that an almost splitting set \(S\) is a \(t\)-complemented \(t\)-splitting set (but not conversely), and that for an almost splitting set \(S\), the (\(t\)-)class group \(\text{Cl}(D)\) is torsion if and only if \(\text{Cl}(D_S)\) and \(\text{Cl}(D_{N(S)})\) are torsion. He also shows that \(D= A+ X^2B[X]\) is an AGCD-domain if and only if \(R= A+ XB[X]\) is an AGCD-domain and \(\text{char}(A)\neq 0\). (Recall that \(A\) is an AGCD-domain if for every \(a, b\in A\), there is an integer \(n= n(a,b)\geq 1\) such that \(a^nA\cap b^nA\) is principal).
reviewed by: David F. Anderson
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 2157529
zbMATH Keywords: almost splitting set
zbMATH Keywords: AGCD-domain
zbMATH Keywords: class group
zbMATH Keywords: integral domain
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.035
OpenAlex ID: W2037273473
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DOI: 10.1016/J.JPAA.2004.08.035
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description: scientific article; zbMATH DE number 2157529