Let $k$ be a field. How can I determine the number of the ideals that the ring $k[x]/(x^2)$ have?
2026-04-07 00:26:38.1775521598
How many ideals does the ring $k[x]/(x^2)$ have?
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Ideals of $k[x]/(x^2)$ are in bijection with ideals of $k[x]$ containing $(x^2)$. An ideal of $k[x]$ contains $(x^2)$ iff it is generated by $p(x)$ such that $p(x)$ divides $x^2$. You should be able to go from here.
EDIT: looks like someone beat me to the answer in the comments. I'm not sure if I should delete this answer?