Two characterizations of pure injective modules

Authors:
Kamran Divaani-Aazar, Mohammad Ali Esmkhani and Massoud Tousi

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

Published electronically:
April 11, 2006

MathSciNet review:
2231603

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Abstract | References | Similar Articles | Additional Information

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.

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Additional 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

Keywords:
Pure injective modules,
injective cogenerators,
finitely embedded modules,
finitely presented modules

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

Article copyright:
© Copyright 2006
American Mathematical Society