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

 

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
Published electronically: April 11, 2006
MathSciNet review: 2231603
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08336-5
PII: S 0002-9939(06)08336-5
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