every PID is a UFD - alternative proof
Proposition. If is a principal ideal domain, then is a unique factorization domain
.
Proof. Recall, that due to Kaplansky Theorem (see this article (http://planetmath.org/EquivalentDefinitionsForUFD) for details) it is enough to show that every nonzero prime ideal in contains a prime element
.
On the other hand, recall that an element is prime if and only if an ideal generated by is nonzero and prime.
Thus, if is a nonzero prime ideal in , then (since is a PID) there exists such that . This completes the proof.