In Dummit and Foote Chapter 15, Section 15.3: Radicals and Affine Varieties on page 679 we find the following definition of affine variety: (see attachment)

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Definition.A nonempty affine algebraic set $\displaystyle V $ is calledirreducibleif it cannot be written as $\displaystyle V = V_1 \cup V_2 $ where $\displaystyle V_1 $ and $\displaystyle V_2 $ are proper algebraic sets in $\displaystyle V $.

An irreducible affine algebraic set is called anaffine variety.

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Dummit and Foote then prove the following results:

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Proposition 17. The Affine algebraic set $\displaystyle V $ is irreducible if and only if $\displaystyle \mathcal{I}(V) $ is a prime ideal.

Corollary 18. The affine algebraic set $\displaystyle V $ is a variety if and only if its coordinate ring $\displaystyle k[V] $ is an integral domain.

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Then in Example 1 on page 681 (see attachment) D&F write:

"Single points in $\displaystyle \mathbb{A}^n $ are affine varieties since their corresponding ideals in $\displaystyle k[A^n] $ are maximal ideals."

I do not follow this reasoning.

Can someone please explain why the fact that ideals in $\displaystyle k[A^n] $ that correspond to single points are maximal

imply that single points in $\displaystyle A^n $ are affine varieties.

Presumably Proposition 17 and Corollary 18 are involved but I cannot see the link.

I would appreciate some help.

Peter