\begin{definition}[Definition:Nagata Criterion]
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 {{iff}} $A$ is a UFD''.
{{NamedforDef|Masayoshi Nagata|cat = Nagata M}}
Category:Definitions/Factorization
\end{definition}