Let(adsbygoogle = window.adsbygoogle || []).push({}); R = Mn(F)the ring consists of all n*n matrices over a field F andE = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix(Eijis a matrix whoseijth 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 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 , 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.REis a maximal left ideal.

2. If A is a rank n-1 matrix inRthen A is equivalent toE, so that there are invertible matrices P and Q such thatA = PEQ. Hence,RA=RPEQ=REQ=REis a maximal left ideal. (note that ifBis invertible andIis a left ideal (resp. a right ideal), thenBI = 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 haveQRE=RE(as stated above) and notREQ=RE. so the red equivalence is incorrect .

So what is the correct proof?

THanks

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# Generator of an ideal , I really need help

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