Rings whose modules are direct sums of extending modules
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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.References
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Additional Information
- Noyan Er
- Affiliation: Department of Mathematics, University of Rio Grande, Rio Grande, Ohio 45674
- Email: noyaner@yahoo.com
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2265-2271
- MSC (2000): Primary 16D10, 16D70, 16P20; Secondary 16G60
- DOI: https://doi.org/10.1090/S0002-9939-09-09807-4
- MathSciNet review: 2495259