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Math Help - question on matrix ring over a field

  1. #1
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    question on matrix ring over a field

    Let
    R = Mn(F) the ring consists of all n*n matrices over a finite field F and

    E
    = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix(Eij is matrix whose ij th element is 1 and the others are 0). Then the following hold:
    1.
    RE is a maximal left ideal.
    2. If
    A is a rank n-1 matrix in R then A is similar to E.

    what is the proof of the above statements?
    thank you

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  2. #2
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    Quote Originally Posted by xixi View Post

    Let R = Mn(F) the ring consists of all n*n matrices over a finite field F and E = E11 + E22 + ... + En-1,n-1,

    where Eii is the elementary matrix(Eij is matrix whose ij th element is 1 and the others are 0). Then the following hold:

    1. RE is a maximal left ideal.
    the result holds for infinite fields as well as finite fields. let I be a left ideal of R which properly contains RE and choose [a_{ij}]=A \in I \setminus RE. since A \notin RE, there must exist 1 \leq k \leq n

    such that a_{kn} \neq 0. see that E_{nn}=a_{kn}^{-1}E_{nk}(A-AE) \in I and thus 1_R=E+E_{nn} \in I. therefore I=R and the proof is complete.
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  3. #3
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    Thank you

    Thank you very much ,you're really the Lord of the Rings ,what do you think of the second statement , mustn't it be "If A is a rank n-1 matrix in RE then A is similar to E "?
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  4. #4
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    Quote Originally Posted by xixi View Post

    what do you think of the second statement , mustn't it be "If A is a rank n-1 matrix in RE then A is similar to E "?
    if A has rank n-1, then one column (we may assume it's the last column) is a linear combination of other columns. then doing some column operations, we can make the entries of the last

    column all zero. so every matrix of rank n-1 is similar to an element of RE. (note that RE is exactly the set of all elements of R whose last columns are zero). so yes, you may assume that

    A \in RE.
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  5. #5
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    how do you prove similarity?

    we should prove that A is similar to E , i.e there must be an invertible matrix P such that PA(P^-1) = E , how would you say this?
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