Decisive dimension and other related torsion theoretic dimensions.
type: Item
instance of: scholarly article
title: Decisive dimension and other related torsion theoretic dimensions.
zbMATH Open document ID: 1113.16035
author: Jaime Castro Pérez
author: Francisco Raggi
author: José Ríos-Montes
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-12T00:00:00Z
review text: A left \(R\)-module \(M\) is called `decisive' if for every hereditary torsion theory \(\tau\), in the category of left \(R\)-modules, \(M\) is either \(\tau\)-torsion or \(\tau\)-torsion-free. In the present paper `the decisive dimension' of a module is defined using the torsion theories which are cogenerated by decisive modules. Section 2 of the paper is dedicated to the study of this dimension. The authors are concerned especially to the question ``when a left \(R\)-module has decisive dimension?''. In Section 3, to each left module \(M\), a class of torsion theories which are cogenerated by decisive modules is associated. Using all these the authors find in Section 4 new characterizations for left semiartinian rings (Corollary 4.2) and for left Artinian rings (Theorem 4.3). I want to mention that the paper contains many interesting and illuminating examples.
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 5124463
zbMATH Keywords: hereditary torsion theories
zbMATH Keywords: decisive modules
zbMATH Keywords: left semi-Artinian rings
zbMATH Keywords: left Artinian rings
zbMATH Keywords: decisive dimension
zbMATH Keywords: categories of left modules
zbMATH Keywords: torsion theoretic dimensions
reviewed by: Simion Sorin Breaz
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.05.026
OpenAlex ID: W1979874487
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DOI: 10.1016/J.JPAA.2006.05.026
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description: scientific article; zbMATH DE number 5124463