# Algebra without Zorn's lemma

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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What is your definition of Noetherian ring? Note that the equivalence of the usual three definitions (every increasing chain stabilizes; every ideal is finitely generated; every nonempty collection of maximal ideals has a maximal element) requires AC/Zorn's Lemma for some of the implications, so which one you pick may be important if you drop AC. –  Arturo Magidin Jan 7 '11 at 21:31
@Arturo This is an interesting point. I guess I am interested in the answer for any of the three definitions you mentioned. –  Vitaly Lorman Jan 7 '11 at 21:36
I've added the tag [axiom-of-choice], I think it's fitting. –  Asaf Karagila Jan 7 '11 at 21:46
Of course, one can do all of finite group theory (which seems to me to go very far) without Zorn's Lemma. –  Arturo Magidin Jan 7 '11 at 21:50