Steadiness of regular semiartinian rings with primitive factors Artinian.
type: Item
instance of: scholarly article
title: Steadiness of regular semiartinian rings with primitive factors Artinian.
zbMATH Open document ID: 1108.16006
author: Jan Žemlička
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: A module \(M\) is called dually slender if the functor \(\Hom(M,-)\) commutes with direct sums; Hyman Bass observed that this is equivalent to requiring that \(M\) is not the union of a countably infinite increasing chain of proper submodules. A ring \(R\) is called (right) steady if the only dually slender right \(R\)-modules are the finitely-generated ones. The main result of this paper is a ring-theoretic characterization of steady rings for a certain class of rings, the regular semiartinian rings \(R\) such that all primitive factor rings of \(R\) are Artinian. The theorem says that such a ring is steady if and only if no factor ring contains an infinitely generated dually slender (right) ideal, and that the characterization is right-left symmetric. The theorem generalizes one in a paper by the author, \textit{P. Růžička} and \textit{J. Trlifaj} [Lect. Notes Pure Appl. Math. 201, 359-371 (1998; Zbl 0917.16004)].
reviewed by: Paul C. Eklof
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 5078837
zbMATH Keywords: unions of countably infinite chains of proper submodules
zbMATH Keywords: direct sums
zbMATH Keywords: finitely generated modules
zbMATH Keywords: dually slender right modules
zbMATH Keywords: right steady rings
zbMATH Keywords: perfect rings
zbMATH Keywords: semiartinian regular rings
Wikidata QID: Q105322672
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.12.011
OpenAlex ID: W2102114040
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DOI: 10.1016/J.JALGEBRA.2005.12.011
description: scientific article; zbMATH DE number 5078837