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.
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