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Math Help - ideals and units

  1. #1
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    ideals and units

    The question is
    Show that if I in M2(R) contains ( 1 0 ), then I=M2(R)
    0 0

    I know the fact that if I contains some unit then it would be equal to the ring R.

    So,
    ( a b ) (1 0 ) = (a 0 )
    c d 0 0 c 0

    (1 0 ) ( A B ) = ( A B
    0 0 C D 0 0

    But how can i say that this is an ideal?
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  2. #2
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    Quote Originally Posted by dreamon View Post
    The question is
    Show that if I in M2(R) contains ( 1 0 ), then I=M2(R)
    0 0

    I know the fact that if I contains some unit then it would be equal to the ring R.

    So,
    ( a b ) (1 0 ) = (a 0 )
    c d 0 0 c 0

    (1 0 ) ( A B ) = ( A B
    0 0 C D 0 0

    But how can i say that this is an ideal?

    \forall\,a,b,c,d\in\mathbb{R} :

    \begin{pmatrix}a&0\\0&0\end{pmatrix}\begin{pmatrix  }1&0\\0&0\end{pmatrix}=\begin{pmatrix}a&0\\0&0\end  {pmatrix}

    \begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix  }0&b\\0&0\end{pmatrix}=\begin{pmatrix}0&b\\0&0\end  {pmatrix}

    \begin{pmatrix}0&0\\c&0\end{pmatrix}\begin{pmatrix  }1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\c&0\end  {pmatrix}

    \begin{pmatrix}0&0\\d&0\end{pmatrix}\begin{pmatrix  }0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&d\end  {pmatrix} -- the right matrix here is in the ideal because of the 2nd equality above .

    Sum all the above up and you'll get that any matrix is in the ideal so...


    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    \forall\,a,b,c,d\in\mathbb{R} :

    \begin{pmatrix}a&0\\0&0\end{pmatrix}\begin{pmatrix  }1&0\\0&0\end{pmatrix}=\begin{pmatrix}a&0\\0&0\end  {pmatrix}

    \begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix  }0&b\\0&0\end{pmatrix}=\begin{pmatrix}0&b\\0&0\end  {pmatrix}

    \begin{pmatrix}0&0\\c&0\end{pmatrix}\begin{pmatrix  }1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\c&0\end  {pmatrix}

    \begin{pmatrix}0&0\\d&0\end{pmatrix}\begin{pmatrix  }0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&d\end  {pmatrix} -- the right matrix here is in the ideal because of the 2nd equality above .

    Sum all the above up and you'll get that any matrix is in the ideal so...


    Tonio
    Dont i have to find
    1 0
    0 1
    as the unit matrix ?
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  4. #4
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    Quote Originally Posted by dreamon View Post
    Dont i have to find
    1 0
    0 1
    as the unit matrix ?

    I don't know what for, but if you insist then just input a=d=1\,,\,b=c=0 in what I posted and we're done.

    Tonio
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