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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 $S$.

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    $\begingroup$ I don't understand your wording. What is the proposition to which you desire a counterexample. For example, could you phrase your question as "Give a counterexample to the following proposition... (proposition here)..." $\endgroup$ – Christian Bueno Jan 8 '14 at 4:02
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$\gcd(2,x)=1$ in $\,\Bbb Z[x],$ but the gcd $\,= 2\,$ in $\,\Bbb Z[x/2]$

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