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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.

user
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    I've read your post and don't have nothing to say. (Eventually I can ask why consider *two* particular fields in the first question as long as the decomposition of that polynomial holds over any field, and add to the second the remark that non-maximal primes of that ring are in fact the minimal ones.) – user26857 May 10 '16 at 13:22
  • Thanks for your comment. For the first question, I was wondering can we identify that ring to any well-known ring? For the second question, is it true that the only maximal prime of that ring is $(x,y)+(xy)$? If possible, please make yours as an answer. I truly want to make this post answered. – user May 10 '16 at 13:48

1 Answers1

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Answering your questions from the comments.

  1. 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 $F[X]/(X^2)\times F[X]/(X^2)$.

  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 $XY$.)

user26857
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    Sorry, I didn't notice that the field is $K=\mathbb C$ in the second question. Then the maximal ideals containing $XY$ are $(X,Y-a)$ and $(X-a,Y)$, $a\in\mathbb C$. – user26857 May 10 '16 at 14:08
  • It's alright. This is exactly what I was looking for. Thanks again. – user May 10 '16 at 14:12