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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rings whose modules are direct sums of extending modules


Author: Noyan Er
Journal: Proc. Amer. Math. Soc. 137 (2009), 2265-2271
MSC (2000): Primary 16D10, 16D70, 16P20; Secondary 16G60
Published electronically: January 30, 2009
MathSciNet review: 2495259
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Abstract: We prove that for a ring $ R$, the following are equivalent: (i) Every right $ R$-module is a direct sum of extending modules, and (ii) $ R$ has finite type and right colocal type (i.e., every indecomposable right $ R$-module has simple socle). Thus, in this case, $ R$ is two-sided Artinian and right serial, and every right $ R$-module is a direct sum of finitely generated uniform modules. This property of a ring is not left-right symmetric. A consequence is the following: $ R$ is Artinian serial if and only if every $ R$-module is a direct sum of extending modules if and only if $ R$ is left serial with every right $ R$-module a direct sum of extending modules.


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Additional Information

Noyan Er
Affiliation: Department of Mathematics, University of Rio Grande, Rio Grande, Ohio 45674
Email: noyaner@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09807-4
PII: S 0002-9939(09)09807-4
Received by editor(s): April 22, 2008
Received by editor(s) in revised form: August 21, 2008, and October 16, 2008
Published electronically: January 30, 2009
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.