Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.

Assume we are given two ideals $I,J$ of finite length in $A$ such that:

- $xJ\subseteq I\subseteq J$

Is it possible to find ideals of finite length $B,C$ in $A$ such that:

- $xC\subseteq B\subseteq C$
- $B\subset I$ and $I/B\cong R/J$
- $C\subset J$ and $J/C\cong R/I$

Here are my thoughts so far: we see that $x$ annihilates $J/I$ so that this is a module of finite length over $\mathbb{C}[[y]]$, hence also some power of $y$ annihilates $J/I$. I thought maybe we should write $I=<f_1,\ldots,f_n>$ and $J=<h_1,\ldots,h_m>$ and see if we can use these generators to construct/find the wanted ideals, but this seems quite hard.

Is it even possible to find such ideals $B$ and $C$? Any ideas how to approach this problem are welcome. One could also ask more genrally:

Assume we are given $n$ ideals $I_r$ of finite length in $A$ such that:

- $xI_r\subseteq I_1\subseteq I_r$ for all $r\geq 2$
- for all $r\geq 2$: $xI_r\subseteq I_s\subseteq I_r$ for all $s\geq r+1$

Is it possible to find ideals of finite length $J_r\subset I_r$ in $A$ for $r=1\ldots n$ which satisfy:

- $xJ_r\subseteq J_1\subseteq J_r$ for all $r\geq 2$
- for all $r\geq 2$: $xJ_r\subseteq J_s\subseteq J_r$ for all $s\geq r+1$
- $I_r/J_r\cong R/I_{r-1}$ ($r\geq 2$) and $I_1/J_1\cong R/I_n$

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