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Math Help - integral domain problem...

  1. #1
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    integral domain problem...

    Q: let R be a ring of 2 by 2 matrices with integer entries.which of the following subset of R is integral domain?
    a) M(0,x,y,0)
    b)M(x,0,0,y)
    c)M(x,0,0,x)
    d)M(x,y,y,z)
    where M stands matrix of elements taken row by row.
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  2. #2
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    Quote Originally Posted by Mathventure View Post
    Q: let R be a ring of 2 by 2 matrices with integer entries.which of the following subset of R is integral domain?
    a) M(0,x,y,0)
    b)M(x,0,0,y)
    c)M(x,0,0,x)
    d)M(x,y,y,z)
    where M stands matrix of elements taken row by row.
    Here you just need to apply the definitions of an integral domain.

    Say that (in first case) \begin{bmatrix}0&a\\b&0\end{bmatrix} \begin{bmatrix}0&c\\d&0\end{bmatrix} = \begin{bmatrix} 0&0\\0&0 \end{bmatrix} so \begin{bmatrix} ad&0\\0&bc \end{bmatrix} = \begin{bmatrix} 0&0\\0&0 \end{bmatrix} which means ad=0,bc=0 and so a=b=c=d=0. So the only way to get the zero matrix is to have zero matrices in the very beginning. Try the same approach for the other three.
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