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 | References | Similar Articles | Additional Information
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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Additional Information
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