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Math Help - Center of a matrix ring

  1. #1
    Junior Member Greg98's Avatar
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    Center of a matrix ring

    Hello,
    the task is to prove that center of a matrix ring R=Mat_2(\mathbb{R}) is the scalar matrices \alpha I, \alpha \in \mathbb{R}.

    I found two more general proofs:
    Center of an Algebra Abstract Nonsense
    The center of a matrix ring over a commutative ring is precisely the scalar matrices Project Crazy Project

    However I didn't understand those totally, so I coudn't reduce them to this simpler case. But I think the main point is:
    Let's assume that M \in Z(Mat_2(\mathbb{R})) and  E=\left(\begin{array}{cc}1&0\\0&0\end{array}\right  ).
    Matrix multiplication ME is commutative ( ME=EM) iff M is in form \left(\begin{array}{cc}a_1&0\\0&a_2\end{array}\rig  ht).

    I don't know how to continue etc., so any help is appreciated. Thanks!
    Last edited by Greg98; March 28th 2011 at 10:59 AM.
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  2. #2
    Junior Member
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    [QUOTE=Greg98;634437]Hello,
    the task is to prove that center of a matrix ring R=Mat_2(\mathbb{R}) is the scalar matrices \alpha I, \alpha \in \mathbb{R}.


    If M=\left(\begin{array}{cc}m_{1}&m_{2}\\m_{3}&m_{4}\  end{array}\right) \in Z(Mat_2(\mathbb{R})) then M \cdot E = E\cdot M, \forall E \in Mat_2(\mathbb{R}).

    If you take  E=\left(\begin{array}{cc}1&0\\0&0\end{array}\right  ) then you have: M \cdot E = \left(\begin{array}{cc}m_{1}&0\\m_{3}&0\end{array}  \right) and E \cdot M= \left(\begin{array}{cc}m_{1}&m_{2}\\0&0\end{array}  \right).

    So, m_{2} = m_{3} = 0.

    If  E=\left(\begin{array}{cc}0&1\\0&0\end{array}\right  ) then you have  m_{1} = m_{4} =m \Rightarrow M=\left(\begin{array}{cc}m&0\\0&m\end{array}\right  ) = m \cdot I.
    Last edited by zoek; March 28th 2011 at 12:48 PM.
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