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Ideals of direct product of rings are direct product of respective ideals?

  1. Sep 27, 2008 #1
    I want to answer this question:
    Find all the ideals of the direct product of rings [tex]R \times S[/tex].
    (I think this means show that the ideals are [tex]I \times J[/tex] where [tex]I, J[/tex] are ideals of [tex]R, S[/tex], respectively.)

    I think the problem is that I don't know how to show that any ideal of [tex]R \times S[/tex] is of the form [tex]A \times B[/tex], where [tex]A \subset R, B \subset S[/tex]. Showing that each are ideals should follow easily enough.

    So I made attemps to prove that [tex](a, m), (b, n) \in K[/tex] iff [tex](a, n), (b, m) \in K[/tex] (where [tex]K[/tex] is an ideal of [tex]R \times S[/tex]), without success...

    can someone help me out?
    thanks in advance.
     
    Last edited: Sep 27, 2008
  2. jcsd
  3. Sep 29, 2008 #2
    never mind. i got it.
    the proposition is false....

    here, i attached the solution.
     

    Attached Files:

  4. Oct 22, 2009 #3
    Hi,
    Is this solution correct?
     
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