Let $M$ be an $A$-module. If every nonempty set of finitely generated submodules of $M$ has a maximal element, then $M$ is Noetherian.

One way to see this is that every submodule of $M$ is finitely generated submodule. Let $N$ be a finitely generated submodule, let $\Sigma$ be a set of all finitely generated submodules of $N$. Then $\Sigma$ has a maximal element, and if it is not $N$, then it contradicts its maximality.QED.

However, one suspicious thing is the assumption that $\Sigma$ is a set. Is $\Sigma$ really a set? In other words, a collection of all finitely generated submodules of a finitely generated module is a set?

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    $\begingroup$ It is a subset of $\mathscr P(M)$. $\endgroup$ – Bernard Jun 25 at 22:00
  • $\begingroup$ @Bernard It's very simple :) Thank you! $\endgroup$ – user124697 Jun 25 at 22:10

$\Sigma$ is a perfectly fine set, as it is a well-defined subset of the power set of $A$.


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