Proving that "Every non-trivial ring (i.e. with more than one element ) with unity has a maximal ideal" implies axiom of choice is true

Question

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 .

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. — Asaf Karagila, Apr 08 '15 at 13:22
I gave three references in my answer, I think that the last one should be accessible without a paywall. — Asaf Karagila, Apr 08 '15 at 13:33

score 5 · Answer 1 · answered Apr 08 '15 at 13:28

5

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 08 '15 at 13:28

Asaf Karagila

405,794

No . the "A New Proof ..." paper also is not open access .. – Apr 08 '15 at 13:35
I said "the last one", and I posted three. You're talking about the second. :-) – Asaf Karagila Apr 08 '15 at 13:39
yeah right :-) . But , is there any open access link to the second? – Apr 08 '15 at 13:41
I don't know. I use a paywall from my university. – Asaf Karagila Apr 08 '15 at 13:44
I'll add that of the three, I definitely prefer the original paper by Hodges. I find it the easiest to digest. – Asaf Karagila Apr 08 '15 at 13:46
@Asaf: And now we know: $\text{some}=1.5$, at least when counting decades. – Brian M. Scott Apr 08 '15 at 17:20
@Brian: The quantifier runs over the real numbers! ;-) – Asaf Karagila Apr 08 '15 at 17:24

Proving that "Every non-trivial ring (i.e. with more than one element ) with unity has a maximal ideal" implies axiom of choice is true

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