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find 3 different prime ideals in $Z[x]$, $I,J,K$ such that $I\subset J\subset K$.

have no clue where to start from.

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Do you know any examples of prime ideals in $\mathbb{Z}[x]$? Can you find a chain of two ideals? –  Qiaochu Yuan Jun 13 '11 at 21:13
    
What is a maximal ideal in $\mathbb{Z}[X]$ –  jspecter Jun 13 '11 at 21:14
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for example $0\subseteq(2)\subseteq(2,x)$ –  yoyo Jun 13 '11 at 21:16
    
Is there a chain of four prime ideals? –  lhf Jun 13 '11 at 21:16
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@jspecter, I meant that question as food for thought for the OP. –  lhf Jun 13 '11 at 21:46

2 Answers 2

Hint 1. Since $\mathbb{Z}[x]$ is an integral domain, there is a very small prime ideal (in fact, it has as few elements as any ideal has any right to have).

Hint 2. There is a one-to-one, inclusion preserving correspondence between the ideals of $R/I$ and the ideals of $R$ that contain $I$. This correspondence preserves primality. So, perhaps you can find some proper prime ideal $P$ such that $\mathbb{Z}[x]/P$ (which must be an integral domain) has a proper prime ideal as well? Then you could "lift" it back to $\mathbb{Z}[x]$, and that would give you your $J$ and $K$. Then use the first hint for your $I$.

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HINT $\ $ Factoring out a prime $\rm\:p \ne 0\:$ reduces it to finding a chain of two prime ideals in $\rm\:\mathbb F_p[x]\:.$

A similar reduction tackles lhf's question in the comments - see Theorem 37 below from Kaplansky's Commutative Rings. (Theorem 34 is the bijective order-preserving correspondence between prime ideals in a localization $\rm\:R_S\:$ and all prime ideal in $\rm\:R\:$ disjoint from $\rm\:S\:$).

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