Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field). We include the empty affine algebraic set among affine algebraic sets and allow the zero ring as a reduced finitely generated $k$-algebra. For a subset $X\subseteq \mathbb A_k^n$, define its vanishing ideal by

\begin{align*} I(X):=\{f\in k[x_1,\dots,x_n]: f(a)=0 \text{ for every } a\in X\}, \end{align*}

and define its coordinate ring by

\begin{align*} k[X]:=k[x_1,\dots,x_n]/I(X). \end{align*}

For an ideal $I\trianglelefteq k[x_1,\dots,x_n]$, define its zero set by

\begin{align*} V(I):=\{a\in \mathbb A_k^n: f(a)=0 \text{ for every } f\in I\}. \end{align*}

The assignment $X\mapsto k[X]$ identifies affine algebraic sets over $k$, up to regular isomorphism, with reduced finitely generated $k$-algebras, up to $k$-algebra isomorphism. More precisely, if $X\subseteq \mathbb A_k^n$ is an affine algebraic set, then $k[X]$ is a reduced finitely generated $k$-algebra. Conversely, if $A$ is a reduced finitely generated $k$-algebra and

\begin{align*} A\cong k[x_1,\dots,x_n]/I \end{align*}

as $k$-algebras for an ideal $I\trianglelefteq k[x_1,\dots,x_n]$, then $I$ is radical and

\begin{align*} A\cong k[V(I)] \end{align*}

as $k$-algebras. Once regular maps are included, this object-level correspondence extends contravariantly: for affine algebraic sets $X\subseteq \mathbb A_k^n$ and $Y\subseteq \mathbb A_k^m$, regular maps $X\to Y$ correspond bijectively to $k$-algebra homomorphisms $k[Y]\to k[X]$.