Definition:Nagata Criterion

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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 if and only if $A$ is a UFD.

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Source of Name

This entry was named for Masayoshi Nagata.