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 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 define

for all "

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

"The following properties of the map are very easy exercises ...

Among these easy exercises is

Despite this being an easy exercise, I cannot see exactly why :-(

Can someone please help?

Peter