Source for problems to practice applying Zorn's lemma.

Question

I have been stuck trying to prove things too many times to mention, where the idea I needed to use was Zorn's lemma. I would like to get to the point where applying transfinite induction is a natural to me as using normal induction. Is there a collection of problems (say on the beginner graduate student level) that use Zorn's lemma?

Answer 1

Here are some problems that uses Zorn's lemma.

1) Every nonzero commutative ring contains a maximal ideal.

2) If every prime ideal in a commutative ring is finitely generated then every ideal in the ring is finitely generated.

3) Every nonzero vector space contains a basis.

You can find more problems here: http://www.math.uconn.edu/~kconrad/blurbs/zorn1.pdf

Answer 2

Since Zorn's lemma is equivalent to the axiom of choice, you can find many examples on the wikipedia page on AC.

Answer 3

You are asking about Zorn's lemma in the title and in the body also about transfinite induction. These are two related topics, but they are not the same. Anyway, you can find a chapter on each of these two topics (including some problems and their solutions) in the book Komjath P., Totik V.: Problems and Theorems in Classical Set Theory, Springer-Verlag, 2006; author's website, Google Books.