Maximal left ideal of matrix ring

  • Thread starter xixi
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let R be a matrix ring over a finite field [tex]\LARGE F_{q}[/tex] , i.e. [tex]\Large R=M_{n}(\LARGE F_{q})[/tex]. then
1.Every matrix of rank n-1 in any maximal left ideal generates the maximal left ideal.
2.moreover,the number of matrices in every maximal left ideal that can be a generator is the same as the number of the generator matrices in the maximal left ideal [tex]\LARGE RE_{11}+...+RE_{n-1,n-1}[/tex] (where [tex]\LARGE E_{ij}[/tex] is n*n matrix whose ij th element is 1 and the others are 0)

what is the proof of the above statements .
Thanks
 

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