Ytukyg

Let $k$ be a field of characteristic zero, and let $R$ be a commutative $k$-algebra, which is perhaps non-affine and/or not an integral domain.
For example: $R=frac{mathbb{Q}[x_1,x_2,dots]}{(x_1x_2)}$.

Is it possible to find $A,B,W in R[X,Y]$ such that: (i) $operatorname{Jac}(A,B)=1$, (ii) $operatorname{Jac}(A,W)=0$, (iii) $W notin R[A]$?

I only succeeded to find elements $A,B,W in R[X,Y]$ satisfying (i) + (iii) or (ii) + (iii), but not satisfying the three conditions.

Several attempts:

• $R=k[t^2,t^3]$, $A=t^2X+t^3Y$, $W=t^3X+t^4Y$ satisfy (ii)+(iii). Indeed, (ii) is clear, $W=tA in k(t^2,t^3)[A]-k[t^2,t^3][A]$, and (i) is not satisfied (since $operatorname{Jac}(A,B)=t^2 E$, for some $E in R[X,Y]$, never equals $1$).




• Taking $R=frac{k[t^2,t^3,s]}{(t^2s=1)}$ does not help, since now $R ni t^3s=t(t^2s)=t1=t$, so although $A$ has a Jacobian mate, $B=sY$ (=(i) is satisfied),
  now (iii) is not satisfied, since $W=tA in R[A]$.




• Taking $R=k((t))$ also does not help. Now $1-t^2$ is invertible, with inverse $epsilon:=1+t^2+t^4+t^6+dots$. $A=(1-t^2)X+(1-t)Y$ has a Jacobian mate $B=epsilon Y$.
  $W=(1+t)X+Y$ satisfies $operatorname{Jac}(A,W)=0$, but $W=(1-t)^{-1}A in R[A]$,
  so (iii) is not satisfied.

Remark: Relying on several considerations that I will not explaim now (perhaps if someone will ask me to explain, then I will explain), it seems that $R$ should be non-affine+not an integral domain, in order to have a chance to get an example with $deg(A),deg(B) < 100$. In other words, it seems that if $R$ is affine or an integral domain, then each of possible $A$ and $B$ must be of degrees $ geq 100$.
Therefore, my above attempts ($R$ is affine) are hopeless.
So we should better consider something like $R=frac{k[[x_1,x_2,dots]]}{(x_1^2)}$.
$A=X+(x_1+x_2+dots)Y$ has a Jacobian mate $B=X+Y$. Can we find an appropriate $W$?
What about $R= frac{k[[x_1,x_1^{-1},x_2,x_2^{-1},dots]]}{I}$, for some appropriate ideal $I$?

Thank you very much!