Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective modules as being those that have no proper essential extension 3) the structure theorem saying that every injective module is the direct sum of injective indecomposable modules.

Question: Is there any profound reason for the necessity of Zorn's Lemma when treating injective modules? Is this somehow related to the fact that there is no dual for the notion of a free module?

Solution 1:

If we only reduce to $\Bbb Z$-modules, then we can already identify a very strong connection:

The axiom of choice is equivalent to the statement "All divisible abelian groups are injective".

Moreover, there is a model where the axiom of choice fails and there are no injective abelian groups, at all, so the above equivalence fails in a very acute way.

You may be interested in some of the work of Andreas Blass, who proved both of the aforementioned results. You can find the paper here:

Blass, Andreas "Injectivity, Projectivity, and the Axiom of Choice." Transactions of the American Mathematical Society, Vol. 255, (Nov., 1979), pp. 31-59

Solution 2:

Note that there is a version of Baer's criterion which is true in ZF:

If $M$ is some $R$-module with the property that for every ideal $I \subseteq R$ the map $\hom(R,M) \to \hom(I,M)$ is surjective, then for every submodule $A \subseteq B$ such that $B/A$ is finitely generated we have that $\hom(B,M) \to \hom(A,M)$ is surjective.

The proof proceeds by induction on a number of elements of $B$ which generate it modulo $A$. More generally, the proof works when there is a generating set of $B/A$ which is indexed by an ordinal number. You need Zorn's Lemma or something alike in order to achieve this.