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Let $M$ be an $R$-module which is equal to a direct sum of simple $R$-submodules:

$M=\bigoplus_{i \in I} S_{i}$ where each $S_{i}$ is simple.

Then if $M$ is Artinian then $M$ is Noetherian. I'm reading the proof and there is a part which says note that the index set must be finite, otherwise if $M=S_{1} \oplus S_{2} \oplus S_{3} ...$ then we can find a descending chain which does not terminate.

Question: isn't this assuming that if $I$ is infinite then $I$ is countable? by writing $M$ as $M=S_{1} \oplus S_{2} \oplus S_{3} ...$ doesn't this implies that $I$ is countable? Why we can assume this? we can always take a countable subset of an infinite set but how can we guarantee that $M$ can still be written as the direct sum of the simple modules which belong to this countable subset?

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up vote 3 down vote accepted

If $I$ is infinite, we can take a countable subset $J\subset I$ and let $M_n=S_{j_n}\oplus S_{j_{n+1}}\oplus \cdots$. Then $M\supset M_1\supset M_2\supset \cdots$ is a descending chain which does not terminate.

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