TITLE: GCD in a subring is GCD in a bigger ring?
QUESTION [5 upvotes]: Let $R$ be a UFD which is a subring of an integral domain $S$. If $r_1$ and $r_2$ are two nonzero elements of $R$ with GCD $d$, is it true that $d$ is also a GCD of $r_1$ and $r_2$ in $S$?
I know this is true if $R$ is a PID.

REPLY [4 votes]: No, e.g. $\rm\:\gcd(2,x) = 1\:$ in $\rm\:\mathbb Z[x]\:$ but the gcd is the nonunit  $\:2\:$ in $\rm\:\mathbb Z[x/2]\subset \mathbb Q[x]$.
But gcds in a PID D persist in extension rings because the gcd may be specified by the solvability of (linear) equations over D   and such solutions always persist in extension rings, see here.