Stability, duality, 2-generated ideals and a canonical decomposition of modules
type: Item
instance of: scholarly article
title: Stability, duality, 2-generated ideals and a canonical decomposition of modules
zbMATH Open document ID: 1072.13506
author: Bruce Olberding
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2005-10-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108567
review text: In this paper the author studies Warfield (and totally divisorial) domains with a notion that is more general than total divisoriality, that of stability. A non-zero ideal of an integral domain is stable if it is projective over its ring of endomorphisms. If every non-zero ideal of \(R\) is stable, then \(R\) is said to be stable. It is well-known that if every ideal of a Macaulay ring can be generated by two elements, then \(R\) is stable [\textit{H. Bass}, Math. Z. 82, 8--28 (1963; Zbl 0112.26604)]. The converse fails, and consequently, even for Noetherian integral domains, stability is weaker than the property that every ideal be 2-generated. The author frames the concept of divisoriality in terms of stability. Since every totally divisorial domain is stable but not vice versa, the class of totally divisorial domains is properly situated between the classes of stable and Warfield domains. In the present paper, the author uses this observation to bridge the two classes of integral domains.
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 2215986
zbMATH Keywords: totally divisorial domain
zbMATH Keywords: stable domains
zbMATH Keywords: Warfield domains
zbMATH Keywords: 2-generated ideals
MaRDI profile type: Publication
cites work: On the ubiquity of Gorenstein rings
cites work: Warfield domains
cites work: Q5569560
cites work: Almost Maximal Integral Domains and Finitely Generated Modules
cites work: Projective summands in generators
cites work: Generalized Dedekind domains and their injective modules
cites work: Q5687544
cites work: Integral domains in which each non‐zero ideal is divisorial
cites work: Dedekind-like behavior of rings with 2-generated ideals
cites work: Stable Ideals and Arf Rings
cites work: Q4052239
cites work: Injective Modules Over Prufer Rings
cites work: Almost maximal prüfer domains
cites work: On the classification of stable domains
cites work: Factorization into Prime and Invertible Ideals
cites work: Globalizing local properties of Prüfer domains
cites work: Modules of injective dimension one over Prüfer domains
cites work: ON THE STRUCTURE OF STABLE DOMAINS
cites work: Two-generated ideals and representations of Abelian groups over valuation rings
cites work: Rings with two-generated ideals
cites work: Stable rings
cites work: Stable rings and a problem of Bass
cites work: Cancellation of modules and groups and stable range of endomorphism rings
Recommended article: Q4434408
Recommended article: Shift modules, strongly stable ideals, and their dualities
Recommended article: On canonical modules of idealizations
Recommended article: Relative Canonical Modules and Some Duality Results
Recommended article: Generalized representation stability and 𝐹𝐼_{𝑑}-modules
Recommended article: Stable degenerations of Cohen-Macaulay modules
Recommended article: Q4640698
Recommended article: Q4615167
Recommended article: I-stable ideals
Recommended article: On the stable Hom relation and stable degenerations of Cohen-Macaulay modules
description: scientific article