I'm studying affine algebraic varieties(or also called closed sets). I'm using the book of Shafarevich. Let's assume that $k=\overline{k}$
There is a proposition that said:
Let $X,Y$ be two closed sets in the affine space $k^n$ , then:
$X,Y$ are isomorphic (i.e there exist a regular map , with a inverse regular map) if and only if:
$K[X],K[Y]$ (they coordinate rings) are isomorphic as k-algebras.
I only proved the first side ( $X,Y$ isomorphic $\Rightarrow$ $K[X],K[Y]$ isomorphic