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 is it true that every left (or both left and right) -module is a direct sum of cyclic modules? Let be a ring in which all idempotents are central. We prove that if is a left Köthe ring (i.e., every left -module is a direct sum of cyclic modules), then is an Artinian principal right ideal ring. Consequently, is a Köthe ring (i.e., each left and each right -module is a direct sum of cyclic modules) if and only if 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.