Example 89.9.1. Let $k$ be a field. The ring $A = k[x, y, z]/(x^ r + y^ s + z^ t)$ is a UFD for $r, s, t$ pairwise coprime integers. Namely, since $x^ r + y^ s + z^ t$ is irreducible $A$ is a domain. The element $z$ is a prime element, i.e., generates a prime ideal in $A$. On the other hand, if $t = 1 + ers$ for some $e$, then
\[ A[1/z] \cong k[x', y', 1/z] \]
where $x' = x/z^{es}$, $y' = y/z^{er}$ and $z = (x')^ r + (y')^ s$. Thus $A[1/z]$ is a localization of a polynomial ring and hence a UFD. It follows from an argument of Nagata that $A$ is a UFD. See Algebra, Lemma 10.120.7. A similar argument can be given if $t$ is not congruent to $1$ modulo $rs$.
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