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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Chain conditions on essential submodules
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by Barbara L. Osofsky PDF
Proc. Amer. Math. Soc. 114 (1992), 11-19 Request permission

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.
References
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • 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