Lemma 10.50.15. Let $A$ be a ring. The following are equivalent

1. $A$ is a valuation ring,

2. $A$ is a local domain and every finitely generated ideal of $A$ is principal.

Proof. Assume $A$ is a valuation ring and let $f_1, \ldots , f_ n \in A$. Choose $i$ such that $v(f_ i)$ is minimal among $v(f_ j)$. Then $(f_ i) = (f_1, \ldots , f_ n)$. Conversely, assume $A$ is a local domain and every finitely generated ideal of $A$ is principal. Pick $f, g \in A$ and write $(f, g) = (h)$. Then $f = ah$ and $g = bh$ and $h = cf + dg$ for some $a, b, c, d \in A$. Thus $ac + bd = 1$ and we see that either $a$ or $b$ is a unit, i.e., either $g/f$ or $f/g$ is an element of $A$. This shows $A$ is a valuation ring by Lemma 10.50.4. $\square$

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