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 is called irreducible if it cannot be written as where and are proper algebraic sets in .
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 is irreducible if and only if is a prime ideal.
Corollary 18. The affine algebraic set is a variety if and only if its coordinate ring is an integral domain.
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Then in Example 1 on page 681 (see attachment) D&F write:
"Single points in are affine varieties since their corresponding ideals in are maximal ideals."
I do not follow this reasoning.
Can someone please explain why the fact that ideals in that correspond to single points are maximal
imply that single points in are affine varieties.
Presumably Proposition 17 and Corollary 18 are involved but I cannot see the link.
I would appreciate some help.
Peter