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?
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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.
Wojowu
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No, because the Bezout identity in $\Bbb Z$ persists in any ring, i.e. $\,ja+kb = 1$ so $\,d\mid a,b\,\Rightarrow\, d\mid 1$
Bill Dubuque
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Unit factors are OK? – Turbo Oct 25 '18 at 20:22
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@Brout It's not clear what you mean. Generally GCDs are defined only up to unit factors. – Bill Dubuque Oct 25 '18 at 20:31