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

Behboodi, M.; Ghorbani, A.; Moradzadeh-Dehkordi, A.; Shojaee, S.

doi:10.1090/S0002-9939-2014-11158-0

On left Köthe rings and a generalization of a Köthe-Cohen-Kaplansky theorem
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by M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S. H. Shojaee PDF

Proc. Amer. Math. Soc. 142 (2014), 2625-2631 Request permission

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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