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## Main Question or Discussion Point

Let

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

what is the proof of above statements ? I really need help , it's so important for me.

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

(hint:

1.

2. If A is a rank n-1 matrix in

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

So what is the correct proof?

THanks

*R = Mn(F)*the ring consists of all n*n 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 thefollowing 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 , it's so important for me.

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. )

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?

THanks