Zorn's lemma and maximal ideals

Question

Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$.

It's easy to prove that Zorn's lemma implies existence of maximal ideals.

I wonder if the converse proposition is true, i. e., does existence of maximal ideals implies Zorn's lemma? My approach was to construct a ring structure on an ordered set which satisfies Zorn's lemma condition such that existence of maximal ideal implies existence of maximal element in the set, but I didn't succeed.

score 5 · Accepted Answer · answered Jul 18 '14 at 16:47

5

Yes, the existence of maximal ideals (in non-trivial rings with unit) implies Zorn's Lemma. A reference is:

Marcel Erné, A primerose path from Krull to Zorn, pdf-link

answered Jul 18 '14 at 16:47

Martin Brandenburg

163,620

Looks familiar. – Asaf Karagila Jul 18 '14 at 18:08
Indeed. I couldn't find it quickly via the search function. – Martin Brandenburg Jul 18 '14 at 19:11
Yeah, it took me more than a few minutes. – Asaf Karagila Jul 18 '14 at 19:24

Zorn's lemma and maximal ideals

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