I have a question about the correspondence theorem of ideals in rings:
Let's say we have $R=k[X]/(X^2)$, $k$ being a field. Then all the ideals $I$ in $R$ have a corresponding ideal $J$ in $k[X]$ that contains $(X^2)$. As I see it, there are three possibilities:
To which ideals in $R$ do they correspond to, though? Having a look at the proof I would say
But this can't be correct, since obviously $(0)$ is an ideal of $k[X]$ which then should also correspond to $(0)$ so the "correspondence" is not 1-1 which it should be. Also the ideal $(0)$ doesn't contain $(X^2)$. Where is my error? Is it simply that the correspondence only works for ideals that contain $(X^2)$? My script is very fishy in this phrasing.