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

  1. #1
    Super Member Bernhard's Avatar
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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:
    ----------------------------------------------------------------------------------------------------------------------------

    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

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


    The proof of (5) above reads as follows:


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

    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

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

    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
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    Last edited by Bernhard; November 9th 2013 at 03:58 PM.
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    Re: Primary Ideals, prime ideals and maximal ideals - D&F Section 15.2

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