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

 

[fresh_42]

Have a look on these papers:
https://arxiv.org/pdf/1711.09289.pdfhttps://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2040618500033335