Let $k$ be a field, let $n \in \mathbb{N}$, and let $X \subset \mathbb{A}^n_k$ be an affine algebraic set. Define its vanishing ideal by $I(X) := \{f \in k[x_1,\ldots,x_n] : f(a)=0 \text{ for every } a \in X\}$, and define its [coordinate ring](/page/Coordinate%20Ring) by $k[X] := k[x_1,\ldots,x_n]/I(X)$. Then $k[X]$ is a reduced ring.