In the book commutative algebra by Atiyah-Macdonald,
On page $75$, it says
Let $N$ be a submodule of $M$, and let $\sum$ be the set of all
finitely generated submodules of $N$. Then $\sum$ is not empty and
therefore has a maximal element.
On page $78$, in exercise $2$, it says
Let $M$ be an $A$-module. If every non-empty set of finitely generated
submodules of $M$ has a maximal element, then $M$ is Noetherian.
Can someone explain what's the difference between two statements and what is going on here?
Why is the first statement something trivial but the second statement something that has to be proved?
Best Answer
Well, as a matter of perspective, both statements are non-trivial and have to be proved.
The first statement is actually deduced from Proposition $6.1$, which establishes a general correspodance between chain and maximal conditions. We just happen to be in a situation (given Noetherianess!) such that this proposition applies. So, the chain condition implies the maximal condition always, and the second statement shows that the maximal condition on finitely-generated submodules only implies the converse already.
What you noticed is very important, in my opinion, as it highlights very well one aspect making Noetherian modules (and rings) special: they come equipped with a version of choice! Let me expand on this.
You know Zorn's Lemma which allows you to construct - whenever certain technical conditions are met - maximal elements of various partially ordered set. One such partially ordered set is the set of finitely-generated submodules of a module. (EDIT: The finitely-generated part is (crucially) needed to satisfy the technical conditions and, in fact, needed for all submodules a priori; as noted by Alex Wertheim there arise problems if not (for example, that supposedly all modules would be Noetherian)).
You do not need Zorn to accomplish the same thing in Noetherian modules! This, of course, boils down to your definition of Noetherian. However, proving that the prominent versions (finitely-generated submodules, ascending chain conditon, maximal condition) are equivalent actually requires some form of choice (as far as I know dependent choice is sufficient but also, more or less, the least you need).
So, while in general you can deduce the first statement using Zorn, you can deduce the second using a different definition of Noetherianess (combined with some less powerful choice) alone.
You might also find my answer to a related post interesting.