Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common divisor of $a,b$ in $S$.
2026-04-09 16:58:52.1775753932
$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring
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$\gcd(2,x)=1$ in $\,\Bbb Z[x],$ but the gcd $\,= 2\,$ in $\,\Bbb Z[x/2]$