Let R = Mn(F) the ring consists of all matrices over a field F and E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix ( Eij is a matrix whose ij th element is 1 and the others are 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 ideal RE11 +· · ·+REn−1,n−1 that 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.
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. )
Since RE is a left ideal so for an invertible matrix Q , we have QRE=RE (as stated above) and not REQ=RE . so the red equivalence is incorrect .
So what is the correct proof?