Prove that $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ is not isomorphic to $\mathbb{C}[x]$

Question

I want to prove that $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ is not isomorphic to $\mathbb{C}[x]$. My attempt is to see that because $\mathbb{C}$ is UFD, then $\mathbb{C}[x]$ is UFD. However, in $\mathbb{C}[x,y,z](y^2z-x^3+xz^2)$, because $y^2z = x(x^2-z^2)$ we have $(y)(y)(z) = (x)(x+z)(x-z)$, hence it is not an UFD and hence the two rings are not isomorphic.

Is this correct? Thanks

Answer 1

The ring $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ is probably not a UFD, but in any case you still have to prove that the elements $x,y,z,x-z,x+z$ are irreducible in $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$, which may not be immediate...

So let me give you an easier argument. The ideal $(x,y)$ of $\mathbb{C}[x,y,z]$ strictly contains $(y^2z-x^3+xz^2)$, so it gives rise to a nonzero ideal in $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$. Moreover, the latter is a nonmaximal prime ideal of $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$, because $\mathbb{C}[x,y,z]/(x,y) \cong \mathbb{C}[z]$ is an integral domain which is not a field.

However, $\mathbb{C}[x]$ is a PID, so in particular every nonzero prime ideal is maximal. Therefore, $\mathbb{C}[x,y,z]/(y^2z-x^3+xz^2)$ and $\mathbb{C}[x]$ cannot be isomorphic.

Answer 2

Observe that $$\left(\mathbb{C}\left[x,y,z\right]/\left(x^{3}-xz^{2}-y^{2}z\right)\right)/\left(x,y,z\right)^{2}\ \simeq\ \mathbb{C}\left[x,y,z\right]/\left(x,y,z\right)^{2}.$$ But the ideal $$\left(x,y,z\right)\ \subseteq\ \mathbb{C}\left[x,y,z\right]/\left(x,y,z\right)^{2}$$ is clearly nonprincipal, so the latter can’t be quotient of any PID, QED.

(Alternatively, think of this argument as computing the dimension of the Zariski cotangent space of $\mathbb{C}\left[x,y,z\right]/\left(x^{3}-xz^{2}-y^{2}z\right)$ at the point $\left(x,y,z\right)\subseteq \mathbb{C}\left[x,y,z\right]/\left(x^{3}-xz^{2}-y^{2}z\right)$.)