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Math Help - Ring

  1. #1
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    Ring

    Hi! I need help with this exercise :


    1)Let R be a ring with 1 .Prove that  (-1)^2=1 in R

    another question , if I have to "u" is a unit in R, "-u" is a unit ? how i prove this ?

    Thanks
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  2. #2
    MHF Contributor

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    Re: Ring

    "-1" is defined, in a ring, as the additive inverse of the multiplicative identity. In particular, 1+ (-1)= 0 so, by the distributive law, (1+ (-1))^2= 1^2+ (1)(-1)+ (-1)(1)+ (-1)^2= 0. Because 1 is the multiplicative identity, 1^2= 1, of course, and 1(-1)= (-1)(1)= -1 so that (1+ (-1))^2= 1- 2+ (-1)^2= -1+ (-1)^2= 0. Since -1 is the additive inverse of 1, and this equation says it is the additive inverse of (-1)^2, it follows that (-1)^2= 1.

    A 'unit' in a ring is a member that has a multiplicative inverse. If u is a unit, there exist "v" such that uv= vu= 1. But we have also that (-u)(-v)= uv= 1 (use the distributive law to prove that) so that the multiplicative inverse of -u is -v.
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