Chain conditions on essential submodules

Osofsky, Barbara L.

doi:10.1090/S0002-9939-1992-1059630-4

Chain conditions on essential submodules
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by Barbara L. Osofsky

Proc. Amer. Math. Soc. 114 (1992), 11-19

DOI: https://doi.org/10.1090/S0002-9939-1992-1059630-4

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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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