I know that assuming axiom of choice or equivalently Zorn's lemma , it can be proved that every non-trivial ring with unity has a maximal ideal (two sided ) . The wiki article on axiom of choice says that this statement regarding existence of maximal ideal in any non-trivial ring with unity is equivalent to axiom of choice , but I am not able to prove this converse implication . Please help . Thanks in advance .
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$\begingroup$ Do you expect to be able to prove everything that is mentioned in Wikipedia? Some proof are difficult, and require more than a handful of preliminary theorems under your belt. $\endgroup$– Asaf Karagila ♦Apr 8, 2015 at 13:22
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$\begingroup$ This has come up before. $\endgroup$– Asaf Karagila ♦Apr 8, 2015 at 13:23
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$\begingroup$ See W. Hodges: Krull implies Zorn $\endgroup$– martiniApr 8, 2015 at 13:27
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$\begingroup$ @AsafKaragila : Can you at least please give a reference ? $\endgroup$– user228168Apr 8, 2015 at 13:27
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$\begingroup$ @martini : I don't have access ... $\endgroup$– user228168Apr 8, 2015 at 13:28
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1 Answer
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The proof is not trivial to come by.
Originally given by Hodges, the proof shows a variant of Zorn's lemma can be proved from the assertion that every commutative ring with a unit has a maximal ideal.
Wilfrid Hodges, Krull implies Zorn, J. London Math. Soc. (2) 19 (1979), no. 2, 285--287.
Some decades later, Banaschewski gave a somewhat different proof of that same fact.
Bernhard Banaschewski, A new proof that “Krull implies Zorn”, Math. Logic Quart. 40 (1994), no. 4, 478--480.
Both paper are not long, and quite readable granted that you're comfortable with reading choice-related papers.
Marcel Erné, A primrose path from Krull to Zorn, Comment. Math. Univ. Carolin. 36 (1995), no. 1, 123–-126.
Is also related to this proof, although I haven't really read that one.
answered Apr 8, 2015 at 13:28
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$\begingroup$ No . the "A New Proof ..." paper also is not open access .. $\endgroup$– user228168Apr 8, 2015 at 13:35
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$\begingroup$ I said "the last one", and I posted three. You're talking about the second. :-) $\endgroup$– Asaf Karagila ♦Apr 8, 2015 at 13:39
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$\begingroup$ yeah right :-) . But , is there any open access link to the second? $\endgroup$– user228168Apr 8, 2015 at 13:41
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$\begingroup$ I don't know. I use a paywall from my university. $\endgroup$– Asaf Karagila ♦Apr 8, 2015 at 13:44
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$\begingroup$ I'll add that of the three, I definitely prefer the original paper by Hodges. I find it the easiest to digest. $\endgroup$– Asaf Karagila ♦Apr 8, 2015 at 13:46