For a field $k$ , we call a set $V \subseteq k^n$ an algebraic set if for some ideal $I $ of $k[X_1,...,X_n]$ ,
$V=Z(I)=\{(a_1,...,a_n)\in k^n : f(a_1,...,a_n)=0, \forall f \in I\}$ .
Define $I(V):=\{f \in k[X_1,...,X_n] : f(a_1,...,a_n)=0,\forall (a_1,...,a_n)\in V\}$ . For an affine algebraic set $V$ in $k^n$ , let
$k[V]:=k[X_1,...,X_n]/I(V)$ . Now let $k=\mathbb C$ , $n=2$. Let $U=Z(XY-1) , V=Z(Y-X^2)$ . Let $f\in \mathbb C[X,Y]$ be an irreducible polynomial of degree $2$ (highest degree of monomial), let $W=Z(f)$ ; then how to show that $\mathbb C[W]$ is isomorphic with either $\mathbb C[U]$or $\mathbb C[V]$ as $\mathbb C$-algebras ?
Due to $\mathbb C$ being algebraically closed, by Hilbert's Nullsetlensatz, we can rephrase the question more algebraically as follows :
Let $f \in \mathbb C[X,Y]$ be an irreducible polynomial of degree $2$ (highest degree of monomial) . Then how to show that
$\mathbb C[X,Y] /(f)$ is isomorphic with either $\mathbb C[X,Y]/(XY-1)$ or $\mathbb C[X,Y]/(Y-X^2)$ as
$\mathbb C$-algebras ?