## Two characterizations of pure injective modules

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- by Kamran Divaani-Aazar, Mohammad Ali Esmkhani and Massoud Tousi
- Proc. Amer. Math. Soc.
**134**(2006), 2817-2822 - DOI: https://doi.org/10.1090/S0002-9939-06-08336-5
- Published electronically: April 11, 2006
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## Abstract:

Let $R$ be a commutative ring with identity and $D$ an $R$-module. It is shown that if $D$ is pure injective, then $D$ is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if $R$ is Noetherian, then $D$ is pure injective if and only if $D$ is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that $D$ is pure injective if and only if there is a family $\{T_\lambda \}_{\lambda \in \Lambda }$ of $R$-algebras which are finitely presented as $R$-modules, such that $D$ is isomorphic to a direct summand of a module of the form $\prod _{\lambda \in \Lambda }E_\lambda$, where for each $\lambda \in \Lambda$, $E_\lambda$ is an injective $T_\lambda$-module.## References

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## Bibliographic Information

**Kamran Divaani-Aazar**- Affiliation: Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran – and – Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran
- Email: kdivaani@ipm.ir
**Mohammad Ali Esmkhani**- Affiliation: Department of Mathematics, Shahid Beheshti University, Tehran, Iran – and – Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran
**Massoud Tousi**- Affiliation: Department of Mathematics, Shahid Beheshti University, Tehran, Iran – and – Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran
- Received by editor(s): December 16, 2004
- Received by editor(s) in revised form: April 21, 2005
- Published electronically: April 11, 2006
- Additional Notes: This research was supported in part by a grant from IPM (No. 83130115)
- Communicated by: Bernd Ulrich
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 2817-2822 - MSC (2000): Primary 13E10, 13C05
- DOI: https://doi.org/10.1090/S0002-9939-06-08336-5
- MathSciNet review: 2231603