Definition:Nagata Criterion

Definition
Let $A$ be a ring.

The Nagata criterion reads as follows:


 * Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. If the localization $A_S$ is a UFD, then so is $A$.

By Localization of UFD is UFD, this is equivalent to:


 * Let $S \subseteq A$ be a multiplicatively closed subset of $A$ generated by prime elements. Then the localization $A_S$ is a UFD $A$ is a UFD.