# Thread: Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

1. ## Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, the set $\displaystyle \mathcal{I} (A)$ is defined in the following text on page 660: (see attachment)

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"While the ideal whose locus determines a particular algebraic set V is not unique, there is a unique largest ideal that determines V, given by the set of all polynomials that vanish on V.

In general, for any subset A of $\displaystyle \mathbb{A}^n$ define

$\displaystyle \mathcal{I}(A) = \{ f \in k[x_1, x_2, ... \ ... \ , x_n ] \ | \ f( a_1, a_2, ... \ ... \ , a_n) = 0$ for all $\displaystyle ( a_1, a_2, ... \ ... \ , a_n) \in A \}$"

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Then at the top of page 661 D&F write: (see attachment)

"The following properties of the map $\displaystyle \mathcal{I}$ are very easy exercises ...

Among these easy exercises is $\displaystyle \mathcal{I}( \mathbb{A}^n ) = 0$

Despite this being an easy exercise, I cannot see exactly why $\displaystyle \mathcal{I}( \mathbb{A}^n ) = 0$ :-(