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Generator of an ideal , I really need help

  1. Apr 13, 2010 #1
    Let 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 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 , 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
     
  2. jcsd
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