The class semigroup of local one-dimensional domains
type: Item
instance of: scholarly article
title: The class semigroup of local one-dimensional domains
zbMATH Open document ID: 1146.13005
author: P. Zanardo
published in: Journal of Pure and Applied Algebra
publication date: 2008-07-10T00:00:00Z
review text: In recent years, several papers have investigated the question of when the class semigroup, derived from the classical definition of the class group, of a commutative integral domain is a Clifford semigroup. This paper investigates conditions for the class semigroup \(S(R)\) of a local one-dimensional domain \((R, \mathfrak m)\) to be Clifford. Recall that a semigroup \(S\) is called Clifford if it satisfies one of the three equivalent conditions: (1) every element of \(S\) is contained in a subgroup \(G\) of \(S\); (2) for every \(x \in S\) there is \(y \in S\) such that \(x = xyx\); (3) \(S\) is a semilattice of groups. Because \textit{S. Bazzoni} [J. Algebra 238, 703--722 (2001; Zbl 1027.13011)] proved that an integrally closed domain has a Clifford class semigroup if and only if the ring is a Prüfer domain of finite character, the author focuses on the case when the ring is not integrally closed. He makes use of Archimedean valuation domains which dominate \(R\). To be specific, he shows that if \(R\) is dominated by a unique (resp., two distinct) Archimedean valuation domain(s) \(V\) (resp., \(V_1\) and \(V_2\)), and if \(S(R)\) is a Clifford semigroup, then \(V\) (resp., \(V_1 \cap V_2\)) coincides with the integral closure \(\overline{R}\) of \(R\), and every element of \(\overline{R}\) has degree less than or equal to two over \(R\). On the other hand, if any of the following conditions hold, then \(S(R)\) is not Clifford: (i) \(R\) is dominated by three distinct Archimedean valuation domains; (ii) \(R\) is dominated by a nondiscrete real valuation \(v\) and \(v(\mathfrak m) >0\); (iii) \(v\) is a real valuation dominating \(R\) and \(v(\mathfrak m) \geq 3\); (iv) \(v\) is a real valuation, dominating \(R\), of a valuation domain \((V, P)\) such that \([V/P:R/\mathfrak m] = 2\) and \(v(\mathfrak m) =2\). When \(R\) contains a nonzero ideal of a DVR \(V\), the author completely determines when \(S(R)\) is Clifford. Another main result of the paper is the foll…
reviewed by: Sandra Spiroff
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5299442
zbMATH Keywords: Clifford class semigroup
zbMATH Keywords: Archimedean valuation domain
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.03.015
OpenAlex ID: W2080454501
cites work: On the biggest maximally generated ideal as the conductor in the blowing up ring
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: On the ubiquity of Gorenstein rings
cites work: Class semigroups of Prüfer domains
cites work: Groups in the class semigroup of a prüfer domain of finite character
cites work: Clifford regular domains
cites work: Groups in the class semigroups of valuation domains
cites work: Strongly Flat Covers
cites work: Almost perfect domains
cites work: On strongly flat modules over integral domains
cites work: On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field
cites work: Fibres formelles d'un anneau local noethérien
cites work: Q4513600
cites work: Ideal semigroups of noetherian domains and Ponizovski decompositions
cites work: The ratliff-rush ideals in a noetherian ring
cites work: Q4126563
cites work: Class semigroups of integral domains
cites work: Stable Ideals and Arf Rings
cites work: Q5735274
cites work: Q3140228
cites work: Stability of ideals and its applications.
cites work: On the classification of stable domains
cites work: ON THE STRUCTURE OF STABLE DOMAINS
cites work: Generic formal fibers and analytically ramified stable rings
cites work: Rings with two-generated ideals
cites work: Q5697157
cites work: Q5483553
cites work: Loewy length of modules over almost perfect domains
cites work: Stable rings and a problem of Bass
cites work: Stable rings
cites work: Q5796078
cites work: Local domains with topologically T-nilpotent radical
cites work: ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS
cites work: The class semigroup of orders in number fields
DOI: 10.1016/J.JPAA.2008.03.015
Recommended article: Class semigroups of integral domains
Recommended article: On the homological classification of semigroups with local units
Recommended article: The class group and local class group of an integral domain
Recommended article: The local \(S\)-class group of an integral domain
Recommended article: Class semigroups of Prüfer domains
Recommended article: Class semigroups and t-class semigroups of integral domains
Recommended article: A classification of one-dimensional local domains based on the invariant \((c-\delta)r-\delta\)
Recommended article: Q4699203
Recommended article: A class of local noetherian domains
Recommended article: Q4527620
description: scientific article; zbMATH DE number 5299442