Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n \ge 0$, and let $X \subset \mathbb A_k^n$ be an affine algebraic set with coordinate ring
\begin{align*}
Y \longmapsto I_X(Y) := \{f \in k[X] : f(y)=0 \text{ for every } y \in Y\}
\end{align*}
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and
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\begin{align*}
\mathfrak a \longmapsto V_X(\mathfrak a) := \{x \in X : f(x)=0 \text{ for every } f \in \mathfrak a\}
\end{align*}
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give an inclusion-reversing bijection between closed algebraic subsets $Y \subseteq X$ and radical ideals $\mathfrak a \trianglelefteq k[X]$. Under this bijection, irreducible closed subvarieties of $X$ correspond exactly to prime ideals of $k[X]$.
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Moreover, the points of $X$ correspond bijectively to the maximal ideals of $k[X]$ by
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\begin{align*}
p \longmapsto \mathfrak m_{p,X}:=\{f\in k[X]: f(p)=0\},
\end{align*}
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and the $k$-algebra of regular functions on $X$ is naturally isomorphic to $k[X]$.
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If $X$ is irreducible, then $k[X]$ is an [integral domain](/page/Integral%20Domain), its function field is