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 localisation $A_S$ is a UFD, then so is $A$

By Localisation 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 localisation $A_S$ is a UFD iff $A$ is a UFD