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

 

On the endofiniteness of a key module over pure semisimple rings


Authors: Nguyen Viet Dung and José Luis García
Journal: Proc. Amer. Math. Soc. 138 (2010), 2269-2278
MSC (2010): Primary 16G10; Secondary 16D70, 16D90
Published electronically: February 23, 2010
MathSciNet review: 2607855
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Abstract: Let $ R$ be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in $ R$-mod, and let $ W$ be the left key $ R$-module; i.e., $ W$ is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left $ R$-modules. We show that if the module $ W$ is endofinite, then $ R$ is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.


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Additional Information

Nguyen Viet Dung
Affiliation: Department of Mathematics, Ohio University - Zanesville, Zanesville, Ohio 43701
Email: nguyend2@ohiou.edu

José Luis García
Affiliation: Department of Mathematics, University of Murcia, 30100 Murcia, Spain
Email: jlgarcia@um.es

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10098-7
PII: S 0002-9939(10)10098-7
Keywords: Pure semisimple ring, ring of finite representation type, preinjective module, endofinite module.
Received by editor(s): February 5, 2009
Received by editor(s) in revised form: June 17, 2009
Published electronically: February 23, 2010
Additional Notes: The second author was supported by the Fundación Séneca of the C.A.R.M
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society