# Definition:Vanishing Ideal of Subset of Affine Space

(Redirected from Definition:Associated Ideal of Subset of Affine Space)

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## Definition

Let $k$ be a field.

Let $n \geq 0$ be a natural number.

Let $k[X_1, \ldots, X_n]$ be the polynomial ring in $n$ variables over $k$.

Let $S \subseteq \mathbb A^n_k$ be a subset of the standard affine space over $k$.

Its **vanishing ideal** is the ideal:

- $I(S) = \{f \in k[X_1, \ldots, X_n] : \forall x \in S : f(x) = 0\}$

## Also known as

The **vanishing ideal** of $S$ is also know as its **associated ideal**.