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.