Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    Is this solution correct?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook