Proof. Let $R$ be a PID. Since every nonzero ideal of $R$ is principal, and $R$ is a UFD (Lemma 10.120.13), this follows from the fact that every irreducible element in $R$ is prime (Lemma 10.120.5) so that factorizations of elements turn into factorizations into primes. $\square$

There are also:

• 9 comment(s) on Section 10.120: Factorization

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).