Affine Varieties - Single Points and maximal ideals

[Math Amateur]

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 V is called irreducible if it cannot be written as V = V_1 \cup V_2 where V_1 and V_2 are proper algebraic sets in V.

An irreducible affine algebraic set is called an affine variety.

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Dummit and Foote then prove the following results:

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

Corollary 18. The affine algebraic set V is a variety if and only if its coordinate ring 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 \mathbb{A}^n are affine varieties since their corresponding ideals in k[A^n] are maximal ideals."

I do not follow this reasoning.

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

imply that single points in 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

 

[R136a1]

Maximal ideals are always prime.

 

[Math Amateur]

Thanks R136a1

Was looking for that relationship in D&F - just found it in D&F ch 7 page 256 ...

Corollary 14: Assume R is commutative. Every maximal ideal of R is a prime ideal

Mind you, it was your post got me looking again :-)

Thanks again.

Peter