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Proceedings of the American Mathematical Society

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



Countable injective modules are sigma injective

Author: Charles Megibben
Journal: Proc. Amer. Math. Soc. 84 (1982), 8-10
MSC: Primary 16A52; Secondary 16A33
MathSciNet review: 633266
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Abstract: In this note we show that a countable injective module is $ \sum $-injective and consequently a ring $ R$ is left noetherian if the category of left $ R$-modules has a countable injective cogenerator. Our proof can be modified to establish the corresponding result for quasi-injective modules. We also give an example of a nonnoetherian commutative ring $ R$ such that the category of $ R$-modules has a countable cogenerator.

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Keywords: Injective module, $ \sum $-injective, cogenerator, noetherian
Article copyright: © Copyright 1982 American Mathematical Society