# Maximal left ideal of matrix ring

1. Mar 4, 2010

### 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