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On left Köthe rings and a generalization of a Köthe-Cohen-Kaplansky theorem


Authors: M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S. H. Shojaee
Journal: Proc. Amer. Math. Soc. 142 (2014), 2625-2631
MSC (2010): Primary 16D10, 16D70, 16P20; Secondary 16N60
DOI: https://doi.org/10.1090/S0002-9939-2014-11158-0
Published electronically: April 22, 2014
MathSciNet review: 3209318
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Abstract: In this paper, we obtain a partial solution to the following question of Köthe: For which rings $ R$ is it true that every left (or both left and right) $ R$-module is a direct sum of cyclic modules? Let $ R$ be a ring in which all idempotents are central. We prove that if $ R$ is a left Köthe ring (i.e., every left $ R$-module is a direct sum of cyclic modules), then $ R$ is an Artinian principal right ideal ring. Consequently, $ R$ is a Köthe ring (i.e., each left and each right $ R$-module is a direct sum of cyclic modules) if and only if $ R$ is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.


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Additional Information

M. Behboodi
Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: mbehbood@cc.iut.ac.ir

A. Ghorbani
Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
Email: a{\textunderscore}ghorbani@cc.iut.ac.ir

A. Moradzadeh-Dehkordi
Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
Email: a.moradzadeh@math.iut.ac.ir

S. H. Shojaee
Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
Email: hshojaee@math.iut.ac.ir

DOI: https://doi.org/10.1090/S0002-9939-2014-11158-0
Keywords: Cyclic modules K\"othe rings, principal ideal rings, uniserial rings
Received by editor(s): June 15, 2010
Received by editor(s) in revised form: March 6, 2011, March 28, 2011, and August 27, 2012
Published electronically: April 22, 2014
Additional Notes: The research of the first author was in part supported by a grant from IPM (No. 89160031).
The first author is the corresponding author
Communicated by: Birge-Huisgen-Zimmermann
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.