Zorn’s lemma, as defined in Wikipedia, is stated as follows:
(Zorn’s lemma) A partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
It can be proved that Zorn’s Lemma is actually equivalent to the Axiom of Choice, thus independent from $ZF$ framework. A proof that relies on Zorn’s Lemma usually goes as follows: For a certain partially ordered set with some property, there is a maximal element of it. Any element $x$ greater than the maximal element does not satisfy that property anymore. By choosing the proper $x$ we can run the proof by contradiction.
I found myself still not used to the classical ways of using Zorn’s Lemma after a while, such as the proof of Alexander Subbase Theorem, and the proof of “Nilradical of a commutative ring is the intersection of all prime ideals.” It seems to me that those proofs are very different than the ordinary proofs I learned before, that is, if I get stuck with coming up a proof of a proposition, I will intentionally construct a maximal element and blindly test if I could reach the result closer. Another problem is that I have no clue at all whether a theorem relies on Zorn’s lemma or not from a first look. Can anyone give some suggestions, that I should practice and learn more applications of Zorn’s Lemma, or, in my opinion, to find a book that systematically give motivations for each proof? Any help would be appreciated. Thank you.