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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:
----------------------------------------------------------------------------------------------------------------------------
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
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