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I found this statement in wikipedia.

A module $M$ is finitely generated if and only if any increasing chain $M_i$ of submodules with union $M$ stabilizes.

This is essentially the same as this question. But the answer there (in the comment) refers to finitely generated objects in a categorical sense. Can someone provide an elementary proof of this statement? (I know how to prove the “$\Longrightarrow$” part, but not the “$\Longleftarrow$” part.)

PS. This is NOT a question about Noetherian Module. We only say that any chain of submodules of $M$: $M_1\subset M_2\subset\cdots$, which satisfies $M = \bigcup_{i=1}^\infty M_i$, will eventually stabilizes.

Expialidocius
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  • The same proof works, word for word. That's why I closed this. – Arturo Magidin Aug 14 '23 at 03:31
  • @ArturoMagidin Sorry to bother you again, sir. I've read your answer there, but still didn't see what you mean by “same proof”. If you mean: if $M$ is not finitely generated, we can find $\langle n_1\rangle \subsetneq \langle n_1, n_2 \rangle \subsetneq \cdots$, but the union of this chain is not necessarily $M$ (when $M$ doesn't have a countable set of generators) and hence this chain doesn't have to stop. So, no contradiction at this stage. Or do you mean some application of Zorn's lemma? – Expialidocius Aug 14 '23 at 04:35
  • If $M$ is finitely generated, all generators are found in some intermediate step. In the converse, go transfinite. The statement doesn't say "increasing sequence", it says "increasing chain". – Arturo Magidin Aug 14 '23 at 05:15

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