It is well known that the axiom of choice can be used to prove Krull's theorem which states that every ring has a maximal ideal. However, i heard once that Krull's theorem is equivalent to the AC (or to Zorn's lemma). Is that true? So, we suggest each ring (perhaps, each commutative ring) has a maximal ideal and now we need to build some ring to prove the AC (Zorn's lemma, Zermelo theorem etc). Could anybody explain how to do that?
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$\begingroup$ Have you tried searching the site before asking? $\endgroup$– Asaf Karagila ♦Commented Apr 24, 2013 at 8:39
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$\begingroup$ mathoverflow.net/questions/27163/… (Also a google search for "axiom of choice"+krull returns that link in the third place, and the paper by Hodges from that answer as the second result). $\endgroup$– Asaf Karagila ♦Commented Apr 24, 2013 at 8:44
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$\begingroup$ Yes, i have, but a bit sluggishly. Thank you! $\endgroup$– NikitaCommented Apr 24, 2013 at 10:01
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$\begingroup$ Nikita, welcome to MathOverflow. Asaf, since the question to which you link to is not the same as this question, why not just answer this question with a link to Hodge's paper? Indeed, I was say that that answer over there is more an answer to this question here than that one (although of course they are closely related). $\endgroup$– Joel David HamkinsCommented Apr 24, 2013 at 12:20
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$\begingroup$ To be honest, after i had asked my question, i found the Hodge's paper at the Journal of London mathematical society, but couldn't subscribe for the Journal (there was something like 'such page doesn't exists'). Also i hope it's not necessary to pay for it. If it's not a problem, can anybody give the link to the Hodge's paper or send it me to the email? $\endgroup$– NikitaCommented Apr 24, 2013 at 12:53
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1 Answer
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Originally the proof was given by Hodges, from 1979 [1].
Banaschewski gave a new proof of the theorem in 1994 [2].
Bibliography:
Hodges, W., Krull implies Zorn. Journal of the London Mathematical Society 2 (1979), 285-287.
Banaschewski, B., A New Proof that “Krull implies Zorn”. Mathematical Logic Quarterly 40 (1994), 478-480.
answered Apr 24, 2013 at 18:54