Maximal submodules and locally perfect rings
type: Item
instance of: scholarly article
title: Maximal submodules and locally perfect rings
zbMATH Open document ID: 0935.16003
DOI: 10.1007/BF02307202
published in: Mathematical Notes
publication date: 2000-02-09T00:00:00Z
review text: A ring is a right Bass ring if all right modules have maximal submodules. Let \(R\) be a ring with center \(C\). The ring \(R\) is locally \(X\) if the localization \(R_M\) has property \(X\) for every maximal ideal \(M\) of \(C\). The main theorem is a long list of equivalent conditions for a locally module-finite ring \(R\) to be a right Bass ring. These include (1) \(R\) is a left Bass ring, (2) \(R\) is locally a right Bass ring, (3) \(R\) is locally a right perfect ring, (4) \(C\) is a locally perfect ring, (5) \(C/J(C)\) is a von Neumann regular ring and \(J(C)\) is left vanishing where \(J(C)\) is the Jacobson radical of \(C\), and (5) \(R\) is a strongly \(\pi\)-regular ring and \(J(R)\) is left vanishing. These results are all correct. However Corollary 4 and Corollary 5 are not correct. In both of these it is asserted that if \(R\) is a locally module-finite right Bass ring which is semiprime, then \(R\) is von Neumann regular. A counterexample can be constructed as follows. Let \(C\) be a commutative von Neumann regular ring with no minimal ideal and \(M\) a maximal ideal. Let \(R\) be the matrix ring of the form \(\left(\begin{smallmatrix} C&C\\ M&C\end{smallmatrix}\right)\). The ring \(R\) is locally module-finite locally perfect and hence is right Bass. Also \(J(R)=0\) but \(R\) is not von Neumann regular.
reviewed by: Robert L. Snider
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P70
Mathematics Subject Classification ID: 16N20
zbMATH DE Number: 1274222
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: right Bass rings
zbMATH Keywords: maximal submodules
zbMATH Keywords: locally module-finite rings
zbMATH Keywords: right perfect rings
zbMATH Keywords: Jacobson radical
author: A. A. Tuganbaev
MaRDI profile type: Publication
cites work: Commutative Rings Over which Every Module has a Maximal Submodule
cites work: Locally perfect commutative rings are those whose modules have maximal submodules
cites work: Q4106702
cites work: Q5831378
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: On the von Neumann regularity of rings with regular prime factor rings
cites work: Central Localizations of Regular Rings
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description: scientific article; zbMATH DE number 1274222