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Math Help - Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

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    Super Member Bernhard's Avatar
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    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  \mathcal{I} (A) is defined in the following text on page 660: (see attachment)

    ------------------------------------------------------------------------------------------------------------

    "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  \mathbb{A}^n define

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

    --------------------------------------------------------------------------------------------------------------

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

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

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

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

    Can someone please help?

    Peter
    Attached Files Attached Files
    Last edited by Bernhard; October 27th 2013 at 03:07 AM.
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