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Math Help - Primary ideals in Z

  1. #1
    Super Member Bernhard's Avatar
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    Primary ideals in Z

    In Dummit and Foote on page 682 Example 1 reads as follows:
    ----------------------------------------------------------------------------------------------------------------------------


    The primary ideals in  \mathbb{Z} are 0 and the ideals  (p^m) for p a prime and  m \ge 1 .


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

    So given what D&F say, (4) is obviously not primary.

    I began trying to show from definition that (4) was not a primary from the definition, but failed to do this

    Can anyone help in this ... and come up with an easy way to show that (4) is not primary?

    Further, can anyone please help me prove that the primary ideals in  \mathbb{Z} are 0 and the ideals  (p^m) for p a prime and  m \ge 1 .


    Peter


    Note: the definition of a primary idea is given in D&F as follows:

    Definition. A proper ideal Q in the commutative ring R is called primary if whenever  ab \in Q and  a \notin Q then  b^n \in Q for some positive integer n.

    Equivalently, if  ab \in Q and  a \notin Q then  b \in rad \ Q
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  2. #2
    Member Haven's Avatar
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    Re: Primary ideals in Z

    (4) is primary, as 4=2^2 and 2 is prime. Suppose ab \in (4). Then 4|ab. Suppose a \not\in (4), so 4\nmid a.

    But we can have two cases, either 2|a or 2\nmid a:

    If 2|a, as 4|ab and 4\nmid a, we get 2|b, which means b\in(2) therefore b^2 \in (4).

    If 2\nmid a, then 4|b, and so b\in 4.

    You can use a similar proof to prove that the primary ideals are (0) and (p^m).
    Thanks from Bernhard
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