I wanted to find an integral domain in which we do not have the property that any two elements have a greatest common divisor (gcd).
Somewhere on this forum someone claimed that the following quotient ring of the polynomial ring in four variables over a field $k$ is such a ring: $A:=k[X,Y,Z,W]/(XW−YZ)$. I would appreciate some help in proving this. We denote the classes of $X,Y,Z,W$ in $A$ by $x,y,z,w$.
1) $A$ is an integral domain
It suffices to prove that $XW−YZ$ is irreducible in $k[X,Y,Z,W]$. But this holds with the Eisenstein criterion if we consider $XW−YZ$ as an element in $k[Y,Z,W][X]$.
2) The elements $xw$ and $xy$ have no gcd in $A$.
Obviously the idea is to use that (because of the relation $xw=yz$), both $x$ and $y$ are common divisors of $xw$ and $xy$. But I'm having a hard time proving rigorously that there cannot exist a gcd.