Ideal class semigroups of overrings
type: Item
instance of: scholarly article
title: Ideal class semigroups of overrings
zbMATH Open document ID: 1121.13022
author: Lucian Sega
published in: Journal of Algebra
publication date: 2007-06-22T00:00:00Z
review text: The ideal class semigroup of an integral domain \(R\), denoted by \(S(R)\), consists of all isomorphism classes of nonzero fractional ideals of \(R \), under usual multiplication. An integral domain \(R\) is called Clifford regular if the ideal class semigroup \(S(R)\) is a Clifford semigroup, i.e., for each nonzero fractional ideal \(I\) of \(R\) there exists a nonzero fractional ideal \(J\) of \(R\) such that \(I = I^2 J\). Clifford regular domains were introduced by \textit{P. Zanardo} and \textit{U. Zannier} [Math. Proc. Camb. Phil. Soc. 115, 379--391 (1994; Zbl 0828.11068)] and \textit{S. Bazzoni} and \textit{L. Salce} [Isr. J. Math. 95, 135--155 (1996; Zbl 0869.13006)]. If \(T\) is an overring of \(R \), there is a canonical semigroup homomorphism between the ideal class semigroup of \(R\) and the ideal class semigroup of \(T\). In the present paper, the author investigates conditions under which this semigroup homomorphism is surjective, in order to apply this information to the study of overrings of Clifford regular domains. Following Olberding's terminology, an integral domain \(R\) is called stable (respectively, finitely stable) if each nonzero (respectively, nonzero finitely generated) fractional ideal \(I\) of \(R\) is invertible in the overring \((I:I)\). \textit{S. Bazzoni} [J. Algebra 238, 703--722 (2001; Zbl 1027.13011)] proved that the class of Clifford regular domains is properly intermediate between the class of stable domains and the class of finitely stable domains; in particular, a noetherian domain is Clifford regular if and only if it is a stable domain. After recovering some known results of Bazzoni, the author proves in certain more general situations that the Clifford regular property is inherited by an overring. In particular, he shows that if \(R\) is a Clifford regular domain such that the integral closure of \( R\) is a fractional overring, then every overring of \(R\) is Clifford regular. He also characterizes among Clifford regular …
reviewed by: Marco Fontana
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13F99
zbMATH DE Number: 5166617
zbMATH Keywords: fractional ideal
zbMATH Keywords: stable domain
zbMATH Keywords: Clifford regular domain
zbMATH Keywords: class semigroup
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.10.030
OpenAlex ID: W2025951432
cites work: Groups in the class semigroups of valuation domains
cites work: Class semigroups of Prüfer domains
cites work: Clifford regular domains
cites work: A Generalization of Zariski's Main Theorem
cites work: Q4513600
cites work: Propinquity of one-dimensional Gorenstein rings
cites work: The ratliff-rush ideals in a noetherian ring
cites work: Well-centered overrings of an integral domain.
cites work: Q5603302
cites work: Q5647004
cites work: Modules over Dedekind-like rings
cites work: Dedekind-like behavior of rings with 2-generated ideals
cites work: Stable Ideals and Arf Rings
cites work: ON THE STRUCTURE OF STABLE DOMAINS
cites work: Q5517787
cites work: Generalized Quotient Rings
cites work: Rings with two-generated ideals
cites work: Two-generated ideals and representations of Abelian groups over valuation rings
cites work: Stable rings and a problem of Bass
cites work: Stable rings
cites work: The class semigroup of orders in number fields
cites work: Q5732721
DOI: 10.1016/J.JALGEBRA.2006.10.030
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description: scientific article