Integer coprime over $\mathbb Z$ but have common factor over an GCD Domain

Question

We known integers can split differently in GCD domains.

Given coprime $a,b\in\mathbb Z$ is it possible that they have a common (possibly unit) factor in a GCD domain?

Answer 1

No. If $a,b$ are coprime in $\mathbb Z$, then for some $x,y\in\mathbb Z$ we have $ax+by=1$. This relation will still hold in any other ring containing $\mathbb Z$, so if $a,b$ had any common nonunit factor, it would divide $1$, which is impossible.

Answer 2

No, because the Bezout identity in $\Bbb Z$ persists in any ring, i.e. $\,ja+kb = 1$ so $\,d\mid a,b\,\Rightarrow\, d\mid 1$

Generally GCDS in a PID persist in any extension domain.