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
Full-text PDF Free Access
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