Affine Varieties - Single Points and maximal ideals

1. Nov 8, 2013

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

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Last edited: Nov 8, 2013
2. Nov 8, 2013

R136a1

Maximal ideals are always prime.

3. Nov 8, 2013

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