Example of GCD in UFD that can't be expressed as linear combination

It is true that in any UFD, any two elements have a GCD (this comes right away from the definition), but the Bezout property (i.e. such a GCD is a linear combination) does not hold in general. Counter-example: in the polynomial ring $\mathbb{Z}[t]$, $2$ and $t$ are coprime, but the ideal $(2,t)$ is strictly contained in the whole ring, whereas it should be equal to it if Bezout were available.

Let's consider $R[x,y]$. The GCD of $x^2y$ and $xy^2$ is $xy$ so we are looking for $a,b\in R[x,y]$ such that $ax^2y+bxy^2=xy\Rightarrow ax+by=1$. But the constant term of both $ax$ and $by$ is $0$, so the constant term of their sum is also zero. Contradiction.

First we have the result that $\mathbb Z$ is a UFD implies $\mathbb Z[x, y, z]$ is a UFD.

$\operatorname{gcd}(xy, xz) = x$ but $x$ cannot be written as a linear combination of $xy$ and $xz$.