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

Then $\displaystyle Q \subseteq M $ so $\displaystyle 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 $\displaystyle M \subseteq rad M $

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

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

Would appreciate some help.

Peter

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

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