I was reading the proof of the fact that every nonzero proper ideal in a dedekink domain factors uniquely into a product of prime ideals. I was stumped by the beautiful application of Zorns Lemma to this prove theorem. It didn't even occur to me slightest that Zorn's Lemma can be used to prove a result of this form.
Now my question is what are the signs one should look for in a statement which may suggest that Zorn's Lemma might be useful to prove that statement? In some cases you can easily see the use of zorn's lemma for example when using it to the prove that a every ring has a maximal ideal. I'm more interested in the non-obvious cases like the one mentioned above.