Solving Ring Theory Question - Centralizer of Division Char(D) ≠ 2

[DukeSteve]

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.

 

[micromass]

You're almost there (remember that $$sx^2=x^2s$$):

$$(sx-xs)x(sx-xs)^{-1}=(sx^2-xsx)(sx-xs)^{-1}=x(xs-sx)(sx-xs)^{-1}=-x$$

 

[DukeSteve]

Ohh thanks a lot I just figured it out!
Please excuse me if I answered before in a rude form, I didn't want to hurt you!

Thank You very much!