## Endomorphism rings of completely pure-injective modules

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- by José L. Gómez Pardo and Pedro A. Guil Asensio
- Proc. Amer. Math. Soc.
**124**(1996), 2301-2309 - DOI: https://doi.org/10.1090/S0002-9939-96-03240-6
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## 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 $S/J$-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, $E/JE$ 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.## References

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## Bibliographic 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
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**124**(1996), 2301-2309 - MSC (1991): Primary 16S50; Secondary 16D50, 16E60, 16P60, 16S90
- DOI: https://doi.org/10.1090/S0002-9939-96-03240-6
- MathSciNet review: 1307555