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Math Help - Degree of a matrix

  1. #1
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    Degree of a matrix

    Let R=M_n(F) where F is a field . Suppose that E_{ij} is a matrix whose (i,j)th entry is 1 and the others are 0 . For any element A\in R , the number of elements of R such as B for which we have RA+RB=R is the degree of A . Now prove that E_{11} has the minimum degree in R-{ 0} .
    Last edited by xixi; June 7th 2010 at 05:08 AM.
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  2. #2
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    Quote Originally Posted by xixi View Post
    Let R=M_n(F) where F is a field . Suppose that E_{ij} is a matrix whose (i,j)th entry is 1 and the others are 0 . For any element A\in R , the number of elements of R such as B for which we have RA+RB=R is the degree of A . Now prove that E_{11} has the minimum degree in R-{ 0} .
    i wanted to ignore this question because, as usual, there's something wrong with your question: if F is an infinite field and B is any unit of R, then RB = R and so RA + RB = R.

    that means the degree, as you defined, of every element of R is infinity (!!) because R will have infinitely many units.

    so my guess is that the + might be \oplus or the degree might be the number of "left ideals" RB, and not the number of "matrices B", such that RA + RB = R.

    by the way, are these problems from a textbook? do you usually change them?
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    Quote Originally Posted by NonCommAlg View Post
    i wanted to ignore this question because, as usual, there's something wrong with your question: if F is an infinite field and B is any unit of R, then RB = R and so RA + RB = R.

    that means the degree, as you defined, of every element of R is infinity (!!) because R will have infinitely many units.

    so my guess is that the + might be \oplus or the degree might be the number of "left ideals" RB, and not the number of "matrices B", such that RA + RB = R.

    by the way, are these problems from a textbook? do you usually change them?
    Yeah , sorry , it wasn't my fault , I thought that I have included all the assumptions but I dropped that F is a finite field .
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