It is fairly known that
$\mathbb C[x,y,z]/(xy+z^n) \cong \mathbb C[x,y,z]/(z^n+x^2+y^2)$.
This appears, for example, in the study of singularities of type $A_n$. But, unfortunately, I am not able to prove it.
How could we establish this isomorphism?
Thanks!