The basic ring of a locally artinian category and applications
type: Item
instance of: scholarly article
title: The basic ring of a locally artinian category and applications
zbMATH Open document ID: 0614.18006
DOI: 10.1007/BF01194295
author: Adalberto Orsatti
author: Claudia Menini
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathcal C}\) be a locally artinian Grothendieck category. The authors associate to \({\mathcal C}^ a \)topological ring (B,\(\beta)\) with the following properties: 1) (B,\(\beta)\) is strictly linearly compact and B/J(B) is a direct product of division rings. 2) The category \({\mathcal C}\) is equivalent to the category \({\mathcal T}_{\beta}\) of all \(\beta\)-torsion left B-modules (this is a slight improvement of a result of \textit{P. N. Ánh} [J. Pure Appl. Algebra 36, 221-224 (1985; Zbl 0563.18006)]. The topological ring (B,\(\beta)\) is called the basic ring of \({\mathcal C}\). (B,\(\beta)\) is canonically associated to \({\mathcal C}\). In fact it is proved that two locally artinian Grothendieck categories are equivalent if and only if their basic rings are topologically isomorphic. Using these results it is proved that if \({\mathcal C}\) has an artinian generator then the basic ring (B,\(\beta)\) of \({\mathcal C}\) is discrete and hence \({\mathcal C}\) is equivalent to B-Mod. This improves a theorem of \textit{C. Năstăsescu} [Commun. Algebra 9, 1981-1996 (1981; Zbl 0481.16006)]. The general Hopkins-Levitzki theorem [obtained by \textit{C. Năstăsescu}, Rev. Roum. Math. Pures Appl. 24, 745-758 (1979; Zbl 0413.18009)] is obtained as a corollary of this result. Finally the class of linearly compact rings whose torsion class has an artinian generator is characterized.
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 3993824
zbMATH Keywords: locally artinian Grothendieck category
zbMATH Keywords: topological ring
zbMATH Keywords: direct product of division rings
zbMATH Keywords: basic ring
zbMATH Keywords: artinian generator
zbMATH Keywords: Hopkins-Levitzki theorem
zbMATH Keywords: linearly compact rings
zbMATH Keywords: torsion class
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01194295
OpenAlex ID: W1977743579
description: scientific article; zbMATH DE number 3993824