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Math Help - Product ideals vs. products of ideals

  1. #1
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    Product ideals vs. products of ideals

    EDIT 1: LaTeX doesn't love me right now, apparently
    EDIT 2: Hooray for not reading super-sticky topics. *embarrassed*

    For one of my problems, I need to show that if I and J are ideals of ring R, then in general \{ ij : i \in I, j \in J \} is not an ideal.
    It asks for a counterexample, but I'm hard-pressed to find one. I'm fairly confident that you can't use principal ideals, which rules out integers, gaussian integers, or integer polynomials, but I feel really uncomfortable at this point with other rings; even subrings of C seem to have really convoluted ideals. That could be my inexperience talking, though.

    The rest of the question asks to show that for that example \sum_v{i_vj_v}, is an ideal; I don't think that will be very hard - it's just the first part that's got me stumped.

    (A later question asks to show that IJ need not be equal to I intersect J, which I think should also fall out of the example above)
    Last edited by Turiski; May 16th 2011 at 07:52 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Turiski View Post
    For one of my problems, I need to show that if I and J are ideals of ring R, then in general \{ ij : i \in I, j \in J \} is not an ideal.

    Choose R=\mathbb{Z}[x],\;I=<x,2>,\;J=<x,3>


    (A later question asks to show that IJ need not be equal to I intersect J, which I think should also fall out of the example above)
    In general IJ\subset I\cap J and IJ=I\cap J\Leftrightarrow I+J=(1)
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