  A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following:
Suppose the Jacobian of $A$ and $B$ is invertible in $\mathbb{C}[x,y]$ and the
Jacobian of $A$ and $w$ is zero for $A,B,w \in \mathbb{C}[x,y]$. Then $w \in
\mathbb{C}[A]$. We show that in CMW's result it is possible to replace
$\mathbb{C}$ by any field of characteristic zero, and we conjecture the
following 'two-dimensional Centralizer Conjecture over $D$': Suppose the
Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and
$w$ is zero for $A,B,w \in D[x,y]$, $D$ is an integral domain of characteristic
zero. Then $w \in D[A]$. We show that if the famous two-dimensional Jacobian
Conjecture is true, then the two-dimensional Centralizer Conjecture is true.
