Let $R$ be a commutative ring with unity. Now assume that $R$ is Unique Factorization Domain, but not necessarily Principal Ideal Domain.
Question: Let $x,y\in R$ be such that their GCD exists in $R$, and let $d=\gcd(x,y)$. Then does there always exists $a,b\in R$ such that $d=ax+by$?
The answer is yes if the ring is PID, but here I am not considering $R$ as a PID, but considering $x,y$ for which GCD exists.
Here $R$ is a UFD, so every element has a factorization. For $x,y\in R$, we say that $d$ is GCD of $x,y$ (provided it exists), if
(1) $d$ divides both $x,y$.
(2)If $c$ divides $x,y$ then $c$ divides $d$.
I think, common divisors of two elements in a UFD always exists (for example, at least $1$). But, GCD does not exists means we can find two divisors, which are maximal but different.