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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Absolutely pure modules


Author: Charles Megibben
Journal: Proc. Amer. Math. Soc. 26 (1970), 561-566
MSC: Primary 16A64
DOI: https://doi.org/10.1090/S0002-9939-1970-0294409-8
MathSciNet review: 0294409
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Abstract: A module $ A$ is shown to be absolutely pure if and only if every finite consistent system of linear equations over $ A$ has a solution in $ A$. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, $ R$ is Noetherian if and only if every absolutely pure $ R$-module is injective and semihereditary if and only if the class of absolutely pure $ R$-modules is closed under homomorphic images. If $ R$ is a Prüfer domain, then the absolutely pure $ R$-modules are the divisible modules and $ \operatorname{Ext} _R^1(M,A) = 0$ whenever $ A$ is divisible and $ M$ is a countably generated torsion-free $ R$-module.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0294409-8
Keywords: Pure submodule, absolutely pure, injective module, Noetherian ring, semihereditary ring, regular ring, Prüfer domain, divisible module, torsion-free module
Article copyright: © Copyright 1970 American Mathematical Society

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