Statements equivalent to the Axiom of Choice

Solution 1:

From the top of my head, some "big" equivalents to the axiom of choice.

1. Every commutative ring with a unity has a maximal ideal.
2. Every surjection has an injective inverse.
3. Every free abelian group is projective.
4. Every divisible abelian group is injective.
5. Skolem-Lowenheim theorems in model theory.
6. Every two cardinals are comparable. This is used often in "hiding" where we have two sets and we know that one necessarily injects into the other.
7. Every spanning set includes a basis (which is stronger than just the existence of a basis).
8. Every set is included in $L[A]$ for some $A$. Where $L[A]$ is a model of $\sf ZFC$. This can be used to prove certain things in set theory.

There are many many many more, and one can scrounge through Rubin & Rubin Equivalents of the Axiom of Choice II to find many of them including proofs.