Let $k$ be a field and we consider the ring $k[x,x^{-1}]\cong k[x,y]/(xy-1)$. It is clear that $k[x,x^{-1}]$ is a UFD (unique factorization domain) but is $k[x,x^{-1}]$ a PID (principal ideal domain)?
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2You can also view it as the localization of $k[x]$ at powers of $x$, whence the duplicate gives you your answer. – rschwieb Aug 08 '22 at 03:52
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2Localization of PID is a PID – MAS Aug 08 '22 at 04:05