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Coxeter Diagrams and the Köthe’s Problem

Part of: Representation theory of rings and algebras Modules, bimodules and ideals

Published online by Cambridge University Press:  24 February 2020

Ziba Fazelpour  and
Alireza Nasr-Isfahani
Ziba Fazelpour
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran e-mail: z.fazelpour@ipm.ir
Alireza Nasr-Isfahani
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran e-mail: nasr_a@sci.ui.ac.irnasr@ipm.ir
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Abstract

A ring $\unicode[STIX]{x1D6EC}$ is called right Köthe if every right $\unicode[STIX]{x1D6EC}$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. We also give a characterization of basic right Köthe rings with radical square zero. Therefore, we give a solution to Köthe’s problem in these two cases.

Keywords

right Köthe ringrepresentation-finite ringspeciespartial Coxeter functorCoxeter functorCoxeter valued quiverseparated diagram

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16G60: Representation type (finite, tame, wild, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 656 - 686
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The research of the first author was in part supported by a grant from IPM. Also, the research of the second author was in part supported by a grant from IPM (No. 98170412).

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