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Proceedings of the American Mathematical Society

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

 

 

Chain conditions on essential submodules


Author: Barbara L. Osofsky
Journal: Proc. Amer. Math. Soc. 114 (1992), 11-19
MSC: Primary 16P70; Secondary 16P20, 16P40
DOI: https://doi.org/10.1090/S0002-9939-1992-1059630-4
MathSciNet review: 1059630
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Abstract: For $ \aleph $ an infinite cardinal and $ M$ a unital right module over a ring $ R$ with 1 or an object in an $ \mathcal{A}\mathcal{B}5$ category, we show that every well ordered ascending (respectively descending) chain of essential submodules of $ M$ has cardinality less than $ \aleph $ if and only if every well ordered ascending (respectively descending) chain of submodules of $ M/\operatorname{socle}\left( M \right)$ has cardinality less than $ \aleph $. We use this to show that a CS module with an $ \aleph $-chain condition on essential submodules is a direct sum of a module with that same chain condition on all submodules plus a semisimple module. Thus a CS module with fewer than $ \aleph $ generators has an $ \aleph $-chain condition on essential submodules if and only if it has that same $ \aleph $-chain condition on all submodules. As an application in the case of an $ {\aleph _0}$-chain condition, we describe the endomorphism ring of a continuous module with acc on essential submodules.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1059630-4
Keywords: Chain conditions, essential submodules
Article copyright: © Copyright 1992 American Mathematical Society