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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.

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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