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Math Help - Ring Theory Question

  1. #1
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    Ring Theory Question

    Hello Experts,

    Here is the question, and what I did:

    Q: Given a ring with division D char(D) != 2, F = Centralizer of D (means that F becomes a field). Given that x in D isn't in F but x^2 is included in F.

    Needed to prove that there exists y in D and y*x*y^(-1) = -x
    and also that y^2 is in C_D({x}) where C_D is the centralizer of the set {x} sub set of D.

    What I did is:

    I know that x is not in F so there exists such s in D that sx!=xs
    Let's call sx-xs = y there is y^-1 because every non zero element in D is invertible.

    Then I just tried to plug it in the equation: (sx-xs)*x*(sx-xs)^(-1) =>
    (sx-xs)^(-1) should be 1/(sx-xs) but it gives nothing.

    Please tell me how to solve it....I know that I miss something, please guide me step by step.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Berkeley, California
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    Quote Originally Posted by DukeSteve View Post
    Hello Experts,

    Here is the question, and what I did:

    Q: Given a ring with division D char(D) != 2, F = Centralizer of D (means that F becomes a field). Given that x in D isn't in F but x^2 is included in F.

    Needed to prove that there exists y in D and y*x*y^(-1) = -x
    and also that y^2 is in C_D({x}) where C_D is the centralizer of the set {x} sub set of D.

    What I did is:

    I know that x is not in F so there exists such s in D that sx!=xs
    Let's call sx-xs = y there is y^-1 because every non zero element in D is invertible.

    Then I just tried to plug it in the equation: (sx-xs)*x*(sx-xs)^(-1) =>
    (sx-xs)^(-1) should be 1/(sx-xs) but it gives nothing.

    Please tell me how to solve it....I know that I miss something, please guide me step by step.
    Firstly, I can't even tell what this is saying. What is the division of the ring?
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