Why is (one of) the criteria for a Noetherian module not implied by Zorn's lemma? 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?
 A: Zorn's lemma tells you that if you consider the set of all submodules, then this set has a maximal element. It does not tell you whether every nonempty collection contains this maximal element, so you cannot conclude that the collection has one.
Now, the collection indeed does have an upper bound (as you've remarked, $ M $ is one such submodule, as it is a submodule of itself) but this upper bound need not be in the collection.
A: 
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.
A: Zorn's lemma says that if $X$ is a poset and every chain has an upper bound then $X$ has a maximal element. It does not say that every chain has a maximal element, which is what's relevant here.
