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Let $R = k[x,y,z]$ and $P = (y^2-xz,x^2y-z^2,x^3-yz)$ ideal of $R$.

  1. Show that $P$ is prime ideal.

  2. Show that $P^2$ is not primary ideal.

    Hint: Show that $(x,y,z)\in \operatorname{Ass}_R(R/P^2)$

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1 Answers1

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1) $P$ is prime because $R/P\cong k[t^3,t^4,t^5]\subset k[t]$.

2) We have $$P^2=(y^2z-xz^2,x^3z-yz^2,y^4-x^2z^2,x^2y^2-yz^2,x^3y-xz^2,x^2yz^2-z^4,x^6-xz^3).$$

So a primary decomposition of $P^2$ is $$P^2=(y^2 -xz,x^2y-z^2,x^3-yz)\cap(z^2,y^2z, x^3z,y^4,x^2y^2,x^3y,x^6).$$ Then $$\operatorname{Ass}_R(R/P^2)=\{(y^2 -xz,x^2y-z^2,x^3-yz),(x,y,z) \}$$ hence $P^2$ is not a primary ideal.

someone
  • 535