TITLE: When can we prove constructively that a ring with unity has a maximal ideal?
QUESTION [17 upvotes]: Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity.  For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd like to do without choice entirely.
So under what additional hypotheses on a ring $R$ can we exhibit one of its maximal ideals while staying in ZF?  (I'd appreciate both hypotheses on the structure of $R$ and hypotheses on what we're given in addition to $R$ itself, e.g. if $R$ is a finitely generated algebra over a field, an explicit choice of generators.)
Edit:  I guess it's also relevant to ask whether there are decidability issues here.

REPLY [3 votes]: You should take a look at Coquand and Lombardi's "A Logical Approach to Abstract Algebra". 
They observe that commutative rings have a purely equational description, and so there are very strong metatheorems that apply to this theory: Birkhoff's completeness theorem for equational logic, of course; and also Barr's theorem, which states that if a geometric sentences is a consequences of a geometric theories with classical logic plus choice, it's also intuitionistically valid. (And all equational theories are also geometric theories.) 
They strengthen Barr's theorem a bit, by characterizing the relevant intuitionistic proofs,  and then "de-Noetherian-ize" several basic theorems which are typically proved using maximal ideals.