Let bars denote passage to \(\mathbb{Q}[x,y,z]/(xy-z^2)\). Prove that \(\bar{P} = (\bar{x}, \bar{z})\) is a prime ideal. Show that \(\bar{xy} \in \bar{P}^2\) but that no power of \(\bar{y}\) lies in \(\bar{P}\). (This shows \(\bar{P}\) is a prime ideal whose square is not a primary ideal.)
Let \(I = (xy-z^2)\). For any coset \(r+I\) (where \(r\) is any representative), define \(\varphi(r+I)\) as \(r\) but with all terms containing \(x\) and \(z\) removed. Note that this map is well defined as for any \(k \in \mathbb{Q}[x,y,z]\), we have \(\varphi(r+k(xy-z^2)) = \varphi(r)\). Furthermore, it can be seen that \(\varphi\) is a ring homomorphism to \(\mathbb{Q}[y]\), an integral domain. Thus \((x+I, z+I) = \ker \varphi\) is a prime ideal.
As for the second part, \(z^2+I\) = \((z^2+xy-z^2)+I\) = \(xy+I\). The latter statement is evident.
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