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Thread: Maximal ideal (x,y) - and then primary ideal (x,y)^n

  1. #1
    Super Member Bernhard's Avatar
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    Maximal ideal (x,y) - and then primary ideal (x,y)^n

    Example (2) on page 682 of Dummit and Foote reads as follows: (see attached)

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

    (2) For any field k, the ideal (x) in k[x,y] is primary since it is a prime ideal.

    For any $\displaystyle n \ge 1 $, the ideal $\displaystyle (x,y)^n $ is primary

    since it is a power of the maximal ideal (x,y)

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

    My first problem with this example is as follows:

    How can we demonstrate the the ideal (x,y) in k[x,y] is maximal



    Then my second problem with the example is as follows:

    How do we rigorously demonstrate that the ideal $\displaystyle (x,y)^n $ is primary.

    D&F say that this is because it is the power of a maximal ideal - but where have they developed that theorem/result?

    The closest result they have to that is the following part of Proposition 19 (top of page 682 - see attachment)

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

    Proposition 19. Let R be a commutative ring with 1

    ... ...

    (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 ideal with rad Q = M

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

    Now if my suspicions are correct and Proposition 19 is being used, then can someone explain (preferably demonstrate formally and rigorously)

    how part (5) of 19 demonstrates that the ideal $\displaystyle (x,y)^n $ is primary on the basis of being a power of a maximal ideal.

    Would appreciate some help

    Peter
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    Last edited by Bernhard; Nov 12th 2013 at 01:34 AM.
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  2. #2
    Super Member Bernhard's Avatar
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    Re: Maximal ideal (x,y) - and then primary ideal (x,y)^n

    I have been doing some reflecting and reading around the two issues/problems mentioned in my post above.

    First problem/issue was as follows:

    "My first problem with this example is as follows:

    How can we demonstrate the the ideal (x,y) in k[x,y] is maximal"

    In the excellent book "Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by David Cox, John Little and Donal O'Shea we find the following theorem (and its proof) on pages 201-202.

    Proposition 9. If k is any field, an ideal $\displaystyle I \subseteq k[x_1, x_2, ... ... , x_n] $ of the form

    $\displaystyle I = (x_1 - a_1, x_2 - a_2, ... ... x_n - a_n) $ where $\displaystyle a_1, a_2, ... ... , a_n \in k $

    is maximal.


    Now (x, y) is of the form mentioned in Cox et al Proposition 9 since $\displaystyle (x,y) = (x-0, y-0) $ and so by Cox et al Proposition 9, (x,y) is maximal

    Can someone confirm that this is correct.

    Now reflecting on my second problem/issue.

    Peter
    Last edited by Bernhard; Nov 12th 2013 at 11:59 AM.
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