Let $A＝\mathbb Q[x,y,z]/(x^2y^2-z^3)$. I want to find all prime but not maximal ideals that include the ideal $(xz)$ in $A$.
I found $(x,z)⊃(xz)$ is the ideal which satisfies the condition. But I cannot find other ideals.
I want to find all of ideals, thank you.
Best Answer
In effect you are seeking prime ideals containing $I=(xz,x^2y^2-z^3)$ in $R=\Bbb Q[x,y,z]$. If $I\subseteq P$, a prime ideal, then $xz\in P$ so either $x\in P$ or $z\in P$.
If $x\in P$ then $P\supseteq(x,xz,x^2y^2-z^3)=(x,z^3)$. The radical of $(x,z^3)$ is $(x,z)$ so $P\supseteq(x,z)$. The ideal $(x,z)$ is the prime ideal you spotted; it's not maximal, but any prime strictly containing it is.
If $z\in P$ then $P\supseteq(z,xz,x^2y^2-z^3)=(x^2y^2,z)$. As $P$ is prime and $x^2y^2\in P$ then either $x\in P$ or $y\in P$. The $x\in P$ case is dealt with, the $y\in P$ case gives $P\supseteq(y,z)$ and the only such $P$ that isn't maximal is $(y,z)$.