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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 for some .
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 for some where M is a maximal idea.
Then so .
... ... 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
Thus either rad M = M (the conclusion D&F use) or rad M = R?
How do we know that ?
Would appreciate some help.
Peter
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