# Maximal left ideal of matrix ring

#### xixi

let R be a matrix ring over a finite field $$\LARGE F_{q}$$ , i.e. $$\Large R=M_{n}(\LARGE F_{q})$$. 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 $$\LARGE RE_{11}+...+RE_{n-1,n-1}$$ (where $$\LARGE E_{ij}$$ 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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#### fresh_42

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"Maximal left ideal of matrix ring"

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