I just saw someone prove the equivalence for a commutative ring $R$:
(1) $R$ is Noetherian;
(2) Every ideal of $R$ is finitely generated;
(3) Every non-empty set of ideals of $R$ has a maximal element.
For $(2) \Rightarrow (3)$, he used the axiom of choice (although Zorn's Lemma would have been more direct). Suppose I would only want to prove $(1) \iff (2)$, could there exists a proof without AC, or is that impossible since (1), (2), (3) are in fact equivalent?
(I have already proven $(2) \Rightarrow (1)$, so in fact my question pertains to $(1) \Rightarrow (2)$.)