Atiyah Macdonald Exercise 6.2

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?

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

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