How do the proper ideals of $\Bbb Q[x]/ \langle x^3 \rangle$ look like?
I know that the elements of it look like $p(x)$ modulo $x^3$. So the ideals of it look like $\langle ax^2+bx+c\rangle $ where $a,b,c \in \Bbb Q$. Now how many of these are distinct? Can I somehow calculate all them out? My book suggests that there are only $3$ of them which are distinct upto isomorphism. What are they?
Please help me in this regard. Thank you very much.
It is a well known result that if $R$ is a ring and $I \triangleleft R$, the ideals of $R/I$ are in bijective correspondence with the ideals of $R$ that contain $I$, with the bijection being taking the image and preimage of sets via the canonical projection.
In our case, $R$ is principal, so the ideals of $\mathbb{Q}[X]/\langle X^3 \rangle$ are in correspondence with the ideals $J \triangleleft \mathbb{Q}[X]$ such that $\langle X^3\rangle \subseteq J$ and since $J$ is a principal ideal, $J = \langle p \rangle$ for some polynomial $p \in \mathbb{Q}[X]$, and thus $\langle X^3\rangle \subseteq J$ if and only if $p | X^3$, that is, if $p \in \{1,X,X^2,X^3\}$.
Therefore, the ideals of $\mathbb{Q}[X]/\langle X^3 \rangle$ are exactly the image of these via the projection. Concretely, they are
$$ 0, \quad \langle \overline{X^2} \rangle, \quad \langle \overline{X} \rangle, \quad \mathbb{Q}[X]/\langle X^3\rangle $$
with all but the last one being proper.
