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Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$. I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by two elements over $\mathbb C[x,y,z]$. How can I do this?

user26857
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adrija
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1 Answers1

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Hint for #1: We know, for example, that $xy\in I$. If $P$ is a prime ideal containing $I$, we can deduce something very useful about $P$.

Hint for #2: Notice that the vector space of degree $2$ polynomials in $I$ is $3$-dimensional, so it cannot be generated by $2$ elements. Use this fact to derive a contradiction if $I$ can be generated as an ideal by $2$ elements.

Andrew Dudzik
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  • Got the first part, thanks. Can't get the second part still, can you please elaborate a bit on your hint? – adrija Sep 25 '14 at 21:13
  • @user173185 If $f$ and $g$ generate $I$, show that the degree $2$ parts of $f$ and $g$ generate the degree $2$ part of $I$ as a vector space. – Andrew Dudzik Sep 25 '14 at 22:34