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Let $R$ be a UFD and $r\in R$ irreducible. If $S=R-(r)$, why is $S^{-1}R$ a PID?

I've finished proving the integral domain part, which was pretty easy.

How to prove it's principal? consider $R \subset S^{-1}R \subset F$, where $F$ is the field of fractions of $R$?

thank you! :p

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