LetR = Mn(F)the ring consists of all $\displaystyle n \times n$ matrices over a fieldFandE = E11 + E22 + ... + En-1,n-1, whereEiiis the elementary matrix( Eij is a matrix whose ij th element is 1 and the othersare 0). Then the following hold:

1.Every matrix of rank n-1 in any maximal left ideal generates the maximal left ideal itself . 2 .Moreover, the number of matrices in every maximal left ideal that can be a generator is the same as the number of matrices in the maximal left idealRE11 +· · ·+REn−1,n−1that can be a generator. 3.Furthermore why any maximal left ideal has a rank n-1 matrix.

what is the proof of above statements ? I really need help .

Here is a hint for it but it seems incorrect as I will explain in the following.

(hint:

1. RE is a maximal left ideal.

2. If A is a rank n-1 matrix in R then A is equivalent to E, so that there are invertible matrices P and Q such that A = PEQ. Hence, RA=RPEQ=REQ=RE is a maximal left ideal. (note that if B is invertible and I is a left ideal (resp. a right ideal), then BI = I (resp. IB = I)).

From 1 and 2 we know that any rank n-1 matrix in R generates a maximal

left ideal. )

SinceREis a left ideal so for an invertible matrixQ, we haveQRE=RE(as stated above) and notREQ=RE . so thered equivalence is incorrect .

So what is the correct proof?

THanks