Primary Ideals, prime ideals and maximal ideals - D&F Section 15.2

• Nov 9th 2013, 04:42 PM
Bernhard
Primary Ideals, prime ideals and maximal ideals - D&F Section 15.2
I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment)

Proposition 19 Part (5) reads as follows:
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Proposition 19.

... ...

(5) Suppose M is a maximal ideal and Q is an ideal with $M^n \subseteq Q \subseteq M$ for some $n \ge 1$.

Then Q is a primary idea, with rad Q = M

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The proof of (5) above reads as follows:

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Proof.

Suppose $M^n \subseteq Q \subseteq M$ for some $n \ge 1$ where M is a maximal idea.

Then $Q \subseteq M$ so $rad \ Q \subseteq rad \ M = M$.

... ... etc

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My problem is as follows:

Why can we be sure that rad M = M?

I know that M is maximal and so no ideal in R can contain M. We also know that $M \subseteq rad M$

Thus either rad M = M (the conclusion D&F use) or rad M = R?

How do we know that $rad \ M \ne R$?

Would appreciate some help.

Peter
• Nov 10th 2013, 01:35 AM