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

 

Endomorphism rings of completely
pure-injective modules


Authors: José L. Gómez Pardo and Pedro A. Guil Asensio
Journal: Proc. Amer. Math. Soc. 124 (1996), 2301-2309
MSC (1991): Primary 16S50; Secondary 16D50, 16E60, 16P60, 16S90
MathSciNet review: 1307555
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $R$ be a ring, $E=E(R_R)$ its injective envelope, $S=% \operatorname {End}(E_R)$ and $J$ the Jacobson radical of $S$. It is shown that if every finitely generated submodule of $E$ embeds in a finitely presented module of projective dimension $\le 1$, then every finitley generated right $S\slash J$-module $X$ is canonically isomorphic to $% \operatorname {Hom}_R(E,X\otimes _S E)$. This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, $E\slash JE$ is completely pure-injective (a property that holds, for example, when the right pure global dimension of $R$ is $\le 1$ and hence when $R$ is a countable ring), then $S$ is semiperfect and $R_R$ is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.


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Additional Information

José L. Gómez Pardo
Affiliation: Departamento de Alxebra, Universidade de Santiago, 15771 Santiago de Compostela, Spain
Email: pardo@zmat.usc.es

Pedro A. Guil Asensio
Affiliation: Departamento de Matematicas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: paguil@fcu.um.es

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03240-6
PII: S 0002-9939(96)03240-6
Received by editor(s): June 23, 1994
Received by editor(s) in revised form: October 5, 1994, and November 29, 1994
Additional Notes: Work partially supported by the DGICYT (PB93-0515, Spain). The first author was also partially supported by the European Community (Contract CHRX-CT93-0091)
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society