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I have difficulties in doing some calculations of heights and Krull dimensions; I hope that somebody could help me unveil the "tricks of the trade". In the following $\alpha,\beta,\gamma$ denote independent variables.

  1. Let the ring map $f:\mathbf{C}[\alpha,\beta]\rightarrow\mathbf{C}[\gamma]$ be defined by $f(\alpha):=\gamma^2$ and $f(\beta):=\gamma^3$ (it is not onto as $\gamma\notin \operatorname{Im}\gamma$) having kernel $(\alpha^3-\beta^2)$ (are you here with me?), which is a prime ideal as the quotient is $\cong$ to a subring of an integral domain. What's the height of this ideal, and what is the Krull dimension of $\mathbf{C}[\alpha,\beta]/(\alpha^3-\beta^2)$?

  2. What are the Krull dims of $\mathbf{C}[\alpha,\beta]/(\alpha^2+\beta,\alpha^3\beta^2)$ and $\mathbf{C}[\alpha,\beta,\gamma]/(\gamma-\alpha\beta)$?

  3. $\mathbf{Z}/n\mathbf{Z}$ ($n>1$)

How would one approach such a problem? Can we find the height by a theorem of Krull? is it a minimal prime? Are the quotients isomorphic to a ring having a known Krull dim? Unfortunately I haven't found similiar problems on the international network.Thanks

Michael Greinecker
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    Height is used for ideals, not for rings. (I don't know for sure if I'm here with you, since that kernel is not clear why coincides to $(\alpha^3-\beta^2)$, but I've proved this on MSE few days ago.) –  Jan 02 '14 at 17:02

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Since $\mathbf{C}[\gamma^2,\gamma^3]\subset \mathbf{C}[\gamma]$ is an integral extension we obtain $\dim\mathbf{C}[\gamma^2,\gamma^3]=\dim\mathbf{C}[\gamma]=?$.

$\mathbf{C}[\alpha,\beta,\gamma]/(\gamma-\alpha\beta)\simeq\mathbf{C}[\alpha,\beta]$ while $\mathbf{C}[\alpha,\beta]/(\alpha^2+\beta,\alpha^3\beta^2)\simeq \mathbf{C}[\alpha]/(\alpha^7)$, so the first dimension is ... and the second is ....

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    I think you know that $R[X]/(X-a)\simeq R$, where $R$ is a commutative ring and $a\in R$. –  Jan 02 '14 at 16:58
  • thank you, this answer is like every answer should be, nice hints and leaves out the details which I like to carry out myself. –  Jan 02 '14 at 17:17
  • I will (I have forgot to accept the other one), but I will first think through every detail of the solution and related questions –  Jan 02 '14 at 17:57
  • alright I understand everything apart from your second isomorphism (and what the Krull dimension of $\mathbf{C}[\alpha]/(\alpha^7)$ is), could you elaborate a bit? as a quotient they are somehow intimidating; I have added another ring, the $\mathbf{Z}/n\mathbf{Z}$. How would we attack this one? and could you also say something about the height of the prime ideal? thanks a lot! –  Jan 02 '14 at 22:44
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  • The ideal is prime and principal, so its height can't be more than one. 2. $\mathbb C[x,y]/(x^2+y)\simeq\mathbb C[x]$ by sending $y$ to $-x^2$ (see also my first comment), so... The dim. of $\mathbb C[x]/(x^7)$ is easy to find if you look at the primes containing $x^7$. How many are there? 3. Same suggestion: look at the primes containing $n\mathbb Z$? How many are there? Can these be contained one into the other?
    • –  Jan 02 '14 at 23:04