I have a problem
Let $R=k[[x,y]]$ and $m$ be the maximal ideal. Let $I$ be a proper ideal of $R$. Then $I$ is isomorphic to an $m$-primary ideal of $R$ as $R$-modules.
I believe it is too beautiful to be true. I don't know how to start.
Thank you for your help
It might be true. If $I$ is a principal ideal, then $I\cong R$ as $R$-modules. You could try to show that every non-principal proper ideal $I$ can be written as $I=fQ$ where $Q$ is $m$-primary.