Krull Implies Zorn

THEOREM. In Zermelo-Fraenkel set theory, the statement " Every unique factorisation domain has a maximal ideal" implies the Axiom of Choice. We begin the proof by paraphrasing the Axiom of Choice. By a tree we mean a partially ordered set (T, ^ ) such that for every te T, the set ? = {reT : r < t) is linearly ordered. A branch in the tree is a maximal linearly ordered subset. Two elements r, t of T are said to be comparable if either r < / or t < r.