I am trying to show that $xR\cap yR$ is not finitely generated where $R=\mathbb{Z}+(x,y)\mathbb{Q}[x,y]$.

Question

My ideals are the following: $xR=x\mathbb{Z}+(x^2,xy)\mathbb{Q}[x,y]$ and $yR=y\mathbb{Z}+(xy,x^2)\mathbb{Q}[x,y]$. So $xR\cap yR=xy\mathbb{Q}[x,y]$. Now it remains to see that $xy\mathbb{Q}[x,y]$ is not finitely generated. And I am stuck here.

Thank you.

Answer 1

Take a finite set $S_i\in xy\mathbb{Q}[x,y]$ let $$S_i=\frac{p_i}{q_i}xy+\cdots$$

and let $q$ be the least common multiple of the $q_i$. Then all elements of $(S_1,S_2\cdots)R$ have coefficients of $xy$ with denominator dividing $q$.