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Math Help - ideal in a matrix ring

  1. #1
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    Smile ideal in a matrix ring

    Hi
    consider the ring of 3x3 matrices with entries in Z36.
    How many different matrices are there in the two sided ideal generated by the matrix diag(0,30,6)?
    Thanks for any help,
    Hedi
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  2. #2
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    Re: ideal in a matrix ring

    You'll have to evaluate the two-sided ideal, meaning left (or right) multiply \begin{pmatrix} 0 & 0 & 0 \\ 0 & 30 & 0 \\ 0 & 0 & 6 \end{pmatrix} by \begin{pmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{pmatrix} with a_i \in \bbmath{Z}_{36}.

    You'll probably get something like \begin{pmatrix} 0 & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & b \end{pmatrix} where a,b \subseteq \bbmath{Z}_{36}. You'll have to figure out what exactly a and b can be

    So, if you get something like a= 0, 10, 20, \text{or } 30 and b = 5, \text{or } 25, then the ideal will have 4 * 2 = 8 elements
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  3. #3
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    Re: ideal in a matrix ring

    Thanks.
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  4. #4
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    Re: ideal in a matrix ring

    Quote Originally Posted by Bingk View Post
    You'll have to evaluate the two-sided ideal, meaning left (or right) multiply \begin{pmatrix} 0 & 0 & 0 \\ 0 & 30 & 0 \\ 0 & 0 & 6 \end{pmatrix} by \begin{pmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{pmatrix} with a_i \in \bbmath{Z}_{36}.

    You'll probably get something like \begin{pmatrix} 0 & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & b \end{pmatrix} where a,b \subseteq \bbmath{Z}_{36}. You'll have to figure out what exactly a and b can be

    So, if you get something like a= 0, 10, 20, \text{or } 30 and b = 5, \text{or } 25, then the ideal will have 4 * 2 = 8 elements
    i think you are vastly underrating the complexity involved here. the matrix ring is not commutative, and has 1,296 elements (its not "small"). furthermore neither AM nor MA where A is an arbitrary matrix in our ring and M is the diagonal matrix diag(0,30,6), will be diagonal.

    what we need are all matrices of the form AM + MB + CMD, where A,B,C,D are arbitrary matrices in our matrix ring over Z36.

    some things that may help:

    30 = -6 (mod 36), so diag(0,20,6) = (6I)*(diag(0,-1,1)). note 6I commutes with the entire ring.

    in Z36, <30> = <6> = {0,6,12,18,24,30}.

    see if you can prove that any entry but the top-left can be any multiple of 6 (in Z36), and that the top-left entry is always 0. this gives us 6*35 = 210 possible members of the ideal.
    Last edited by Deveno; October 22nd 2012 at 10:17 AM.
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    Re: ideal in a matrix ring

    The general form of a matrix in the two sided ideal involved products of parameters,so the counting is not so simple.In Z36 2x6=8x6
    for example,so different products may be equal in Z36.
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  6. #6
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    Re: ideal in a matrix ring

    it appears i made a mistake, because (as was shown in a later thread) E11 can be written as AMB.

    this shows that our ideal is 6R.

    i also badly calculated the size of the ring, and the ideal:

    R has 369, elements, therefore the ideal has 69 elements.
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  7. #7
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    Re: ideal in a matrix ring

    Yes,this is the answer i ultimately got.
    Thank's a lot.
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