= Nagata criterion for unique factorization domains
{c}
Let $A$ be an <integral domain> that is a <Noetherian ring>, and let $x\in A$ generate a <prime ideal>. If the <localization of a ring> $A_x$ is a <unique factorization domain>, then $A$ is a unique factorization domain. Indeed, integrality of $a/x^n$ over $A$, with $n$ minimal, forces $x\mid a$ and hence proves that $A$ is an <integrally closed domain>. The <Nagata theorem for divisor class groups> then says that $\operatorname{Cl}(A)$ is generated by the height-one primes containing $x$; the only one is the principal prime $(x)$, so $\operatorname{Cl}(A)=0$.
Back to article page
