Let $k$ be a field, let $n\in\mathbb N$, and let $\mathbb A_k^n$ denote affine $n$-space over $k$, whose points are $n$-tuples $a=(a_1,\dots,a_n)\in k^n$. For every subset $X\subset \mathbb A_k^n$, the vanishing ideal
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\begin{align*}
I(X):=\{f\in k[x_1,\dots,x_n]: f(a)=0 \text{ for every } a\in X\}
\end{align*}
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is an ideal of the [polynomial ring](/page/Polynomial%20Ring) $k[x_1,\dots,x_n]$.