Commutative torsion stable rings
type: Item
instance of: scholarly article
title: Commutative torsion stable rings
zbMATH Open document ID: 0543.13009
DOI: 10.1016/0022-4049(84)90011-2
author: Jay Shapiro
author: Robert F. Damiano
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: All rings are commutative with identity. A ring R is called stable provided \(Hom(X,E)\neq 0,\) for some R-module X and injective R-module E, implies \(Hom(Y,E)\neq 0\) for any essential submodule Y of X. For example, Noetherian rings and perfect rings are stable but examples are given of stable rings which are neither Noetherian nor perfect. It is shown that some properties of Noetherian rings can be extended to stable rings. For example, any finitely generated ideal of a stable ring has the AR property, and thus if the polynomial ring R[x] is stable then R is Noetherian (theorem 3.2). It is proved that if R is a stable ring with finite global and Gabriel dimensions such that every semiprime ideal is a finite intersection of prime ideals then R is Noetherian. On the other hand, if R is stable, coherent and has Gabriel dimension then R is Noetherian. The authors ask whether every semiprime ideal is a finite intersection of prime ideals for any stable ring R. \(\{\) Reviewer's comment: Theorem 3.2 is an immediate consequence of proposition 1.5 because it is easy to show that if (x) has the AR property in the polynomial ring R[x] then R is Noetherian.\(\}\)
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13D05
zbMATH DE Number: 3863333
zbMATH Keywords: Noetherianness
zbMATH Keywords: stable ring
zbMATH Keywords: finite dimension
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90011-2
OpenAlex ID: W2086905998
cites work: On the Dimension of Modules and Algebras (III): Global Dimension
cites work: On consequences of stability
cites work: Q4055809
cites work: The Gabriel dimension of a module
cites work: Schranken für die Gabriel- und Krull-Dimension Idealisator-ähnlicher Ringerweiterungen
cites work: Q5603302
cites work: A generalization of quasi-Frobenius rings
cites work: On Stable Noetherian Rings
cites work: Quotient Rings of a Ring and a Subring Which Have a Common Right Ideal
cites work: Q5656860
cites work: The Dual of the Notion of “Finitely Generated”
description: scientific article; zbMATH DE number 3863333