One can't get too far in abstract algebra before encountering Zorn's Lemma. For example, it is used in the proof that every nonzero ring has a maximal ideal. However, it seems that if we restrict our focus to Noetherian rings, we can often avoid Zorn's lemma. How far could a development of the theory for just Noetherian rings go? When do non-Noetherian rings come up in an essential way for which there is no Noetherian analog? For example, Artin's proof that every field has an algebraic closure uses Zorn's lemma. Is there a proof of this theorem (or some Zorn-less version of this theorem) that avoids it?
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The proof of existence and uniqueness of algebraic closures goes through assuming only the ultrafilter lemma, which is strictly weaker than AC; see this MO question. Exactly how strong this assumption is relative to other well-known forms of AC appears to be unknown. I don't know what "Noetherian version of this theorem" means, since every field is Noetherian.