Is it always true that in an integral domain $R$, if $x,y\in R$ and $x,y$ coprime then $ux+vy=1$ for some $u,v\in R$?
I know the converse is true. I don't know if this direction is true.
If the statement is true. Given an integral domain $R$ and let $x,y \in R$. We let $h$ to be the hcf of $x$ and $y$. Then $x=ha$, $y=hb$. $a,b$ are coprime. Hence $ua+vb=1$ and $ux+vy=h$. Then $R$ is a Bezout domain.
It seems like only when $R$ is a Bezout domain, the proposition is true.