TITLE: Why is (one of) the criteria for a Noetherian module not implied by Zorn's lemma?
QUESTION [6 upvotes]: One of the equivalent definitions of a Noetherian $R$-module is

Every nonempty collection of $R$-submodules has a maximal element under inclusion

so, what I'm wondering is, why is this not an immediate application of Zorn's lemma that applies to any module? Our $R$-submodules have a natural partial order under inclusion, and they are bounded above by $M$ because they all must be contained in $M$.
Of course my logic must be wrong somewhere because not every module is Noetherian. What am I missing?

REPLY [3 votes]: Zorn's lemma If every chain in a partially ordered set has an upper bound, then the partially ordered set has a maximal element.

The condition you are mentioning is about any (non empty) collection of submodules having a maximal element. Such collections need not satisfy the condition that every chain has an upper bound, so Zorn's lemma cannot apply.
It's easy to show an example of a partially ordered set (indeed a totally ordered set) that doesn't satisfy the hypothesis of Zorn's lemma, namely the natural numbers under the usual ordering. Note that no infinite subset of the natural numbers has a maximal element.
Turning back to modules, some (non empty) collections of submodules have maximal elements: the finite collections and also all collections containing the module itself.
However, there are examples of module such that no infinite set of proper submodules has a maximal element. One of them is the Prüfer $p$-group, where the proper subgroups are linearly ordered just like the natural numbers.