# Definition:Vanishing Ideal of Subset of Affine Space

## Definition

Let $k$ be a field.

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

Let $k \sqbrk {X_1, \ldots, X_n}$ 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:

$\map I S = \set {f \in k \sqbrk {X_1, \ldots, X_n} : \forall x \in S : \map f x = 0}$

## Also known as

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