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Let $K$ be an algebraically closed field, why $K[x,y]$ and $K[x,y,z]/(xz-y^2)$ are not isomorphic as rings?
$K[x,y]$ is a regular ring (all of its localizations are regular). The local ring of $K[x,y,z]/(xz-y^2)$ at the origin $(0,0,0)$ is not regular: The maximal ideal $\mathfrak m$ there cannot be generated by two elements, not even mod $\mathfrak m^2$.
Required, but never shown