When the ring of regular functions is a UFD?

Question

Let $X$ be an irreducible affine variety over $\mathbb{k}$. There is the following theorem in algebraic geometry: the algebra $\mathbb{k}[X]$ of regular functions is a UFD if and only if each irreducible submanifold of $X$ of codimension $1$ is a set $V(f)$ of zeros of some regular $f\in\mathbb{k}[X]$. I can imagine the proof and it does not seem to be complicated. However I guess that it can be derived from some results from commutative algebra (something like Krull's theorem, isn't it?). Could you give me a reference for the general theorem?

Answer 1

The relevant theorem in commutative algebra is:

Let $A$ be a noetherian domain. The ring $A$ is a UFD iff every height one prime ideal is principal.

This is Theorem 20.1, page 161, in Matsumura's Commutative Ring Theory .