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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:

  1. $J_1=(X^2)$
  2. $J_2=(X)$
  3. $J_3=k[X]$

To which ideals in $R$ do they correspond to, though? Having a look at the proof I would say

  1. $I_1=J_1/(X^2)=(0)$
  2. $I_2=J_2/(X^2)=(X)$
  3. $I_3=J_3/(X^2)=R$

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.

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  • 3
    $\begingroup$ The theorem is about the correspondence between the ideals of $k[X]/(X^2)$ and the ideals of $k[X]$ that contain $(X^2)$. So in your case, $(0)$ (as an ideal of $k[X]$) doesn't qualify as an ideal that contains $(X^2)$. $\endgroup$ – gniourf_gniourf Jan 1 '18 at 12:01
  • $\begingroup$ Okay, I understand. Are my $I_{1,2,3}$ correct still? $\endgroup$ – Buh Jan 1 '18 at 12:07
  • $\begingroup$ They look good, yes! $\endgroup$ – gniourf_gniourf Jan 1 '18 at 12:08

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