TITLE: When to use Zorn's Lemma
QUESTION [13 upvotes]: I was looking at an exercise this morning which I was able to reduce to showing that the nilradical is the the intersection of the prime ideals in a ring -- a fact I remembered was true, but which I tried for a while to prove without success.  Bucking my usual tendency to let something like that ruin the rest of my day for research, I dug out my copy of Atiyah and MacDonald and looked up the answer.  (The idea is to assume that some non-nilpotent element f lies in every prime ideal, apply Zorn's lemma to the ideals which contain no power of f ordered by inclusion, and then show that the upper bound is prime.)
My reaction to this was something along the lines of, "Ah, I never would have got that, because I never would have tried using Zorn's lemma!"  Upon further reflection, I realized that this indicated a serious weakness in my ability to do commutative algebra.
I'm perfectly comfortable using Zorn's lemma for something like showing that an arbitrary vector space has a basis, but when I look at a question like this I'm just not seeing the connection.  I know that this doesn't really have a definite answer, but I was hoping that someone would be able to point out some kind of connection that would improve my intuition for when Zorn's lemma might be effective.
EDIT: Thanks to everyone for all the answers.  They are all helpful and I had a hard time choosing!

REPLY [4 votes]: IMO, you rarely set out to do a problem and think "Oh, I'll use Zorn's lemma! If I can only figure out how...." Instead, the opportunity comes up naturally, and you just have to notice it.
For example, the proof that every vector space has a basis; everyone learns in their first linear algebra course the algorithm to prove a finite-dimensional vector space has a basis:

Start with the empty set
If your current set is a basis, you're done
Pick some element not in the span of your current set
Add that element to your current set
Go to step 2

Since the vector space is finite dimensional, this algorithm stops and you get a basis.
Now, to prove that all vector spaces have bases, you don't have to do anything new -- you already know the key idea, and in this case you merely have to notice that this algorithm still terminates (but after transfinitely many steps).
Of course, I tend to invoke the well-ordering theorem in my proofs rather than Zorn's Lemma -- as you might guess from my example -- but the same idea can apply.
For the problem in your opening post, the key idea seems to be

Choose an ideal that doesn't contain a power of $x$

And to notice that any particular violation of primeness can be corrected by picking a bigger ideal. It's at this point you notice "Hey, Zorn's lemma lets me do that!" and invoke it to complete your proof.