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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 [itex] M^n \subseteq Q \subseteq M [/itex] for some [itex] n \ge 1 [/itex].

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 [itex] M^n \subseteq Q \subseteq M [/itex] for some [itex] n \ge 1 [/itex] where M is a maximal idea.

Then [itex] Q \subseteq M [/itex] so [itex] rad \ Q \subseteq rad \ M = M [/itex].

... ... 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 [itex] M \subseteq rad \ M [/itex]

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

How do we know that [itex] rad \ M \ne R [/itex]?

Would appreciate some help.

Peter

Proposition 19 Part (5) reads as follows:

----------------------------------------------------------------------------------------------------------------------------

Proposition 19.

... ...

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

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

----------------------------------------------------------------------------------------------

The proof of (5) above reads as follows:

----------------------------------------------------------------------------------------------

Proof.

Suppose [itex] M^n \subseteq Q \subseteq M [/itex] for some [itex] n \ge 1 [/itex] where M is a maximal idea.

Then [itex] Q \subseteq M [/itex] so [itex] rad \ Q \subseteq rad \ M = M [/itex].

... ... etc

----------------------------------------------------------------------------------------------

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 [itex] M \subseteq rad \ M [/itex]

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

How do we know that [itex] rad \ M \ne R [/itex]?

Would appreciate some help.

Peter