Iff propositions where both directions require choice?

Recently, I have been revising a basic course on noncommutative rings and modules over them. One proposition proven early on is all left modules over $$R$$ are free iff $$R$$ is a division ring and an interesting point is that, at least in the standard proofs we were given (and I have seen repeated in various places across this site), both directions use the axiom of choice!

As a quick outline, if $$R$$ is a division ring then we use Zorn's lemma to find a maximal linearly independent set in any left $$R$$-module and this is a basis. If $$R$$ is a ring such that every left $$R$$-module is free, then we use Zorn to find a maximal left ideal in $$R$$ and then $$R/I$$ is a simple left $$R$$-module so $$\text{End}_R(R / I)$$ is a division ring and then you show it is isomorphic to $$R$$ with an explicit isomorphism.

My question is: what are some other interesting examples of "iffs" in mathematics where (at least in the standard proofs or otherwise) both implications use choice? Are there other examples in algebra?