Review on polynomial quotient rings: $F[x]/(x^{4}-2x^{2}+1)$ & $\Bbb C[x,y]/(xy)$ I have searched problems about quotient rings on our site. I think I now have a certain understanding about problem like

(1) Find all ideals of $F[x]/(x^{4}-2x^{2}+1)$ when $F=\Bbb C, \Bbb R.$
(2) Find all prime ideals which are not maximal of $\Bbb C[x,y]/(xy)$.

I will show my work here and hopefully get your feedback.
For (1), I found related and helpful answers here and there. I therefore can say
there are $9$ ideals of $F[x]/(x^{4}-2x^{2}+1)$ which are $I/(x^{4}-2x^{2}+1)$ where
$I=((x-1)^i(x+1)^j), i,j \in \{0,1,2\}$. The key here is that $F[x]$ is PID.
For (2), prime ideals of $\Bbb C[x,y]/(xy)$ are precisely prime ideals of  $\Bbb C[x,y]$ that
contain $(xy)$. We want to find the prime ideals that are not maximal.
This post tells me these ideals are principal ideals $(f(x,y))$ where $f$ is irreducible.
In particular, $xy \in (f)$ we must have either $f=x$ or $f=y$ (excluded $f=1$).
As a consequence, ideals that we wanted to find are $(x)+(xy)$ and $(y)+(xy)$.
These are indeed prime by this question and answer.
Thanks for reading my post.

Answering your questions from the comments.


*

*We have $$F[X]/((X-1)^2(X+1)^2)\simeq F[X]/((X-1)^2)\times F[X]/((X+1)^2)$$ by CRT. Moreover,  the last one is isomorphic to $F[X]/(X^2)\times F[X]/(X^2)$.

*There are many more maximal ideals containing $XY$ than $(X,Y)$. Notice that $(XY)\subset (X)\subset (X,f(Y))$ with $f\in K[Y]$ irreducible, and also $(XY)\subset (Y)\subset (Y,f(X))$ with $f\in K[X]$ irreducible. (In fact, these are all maximal ideals containing $XY$.)
