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Math Help - finite ring with an odd number of invertible elements

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    finite ring with an odd number of invertible elements

    Suppose R is a finite ring with an odd number of invertible elements. Prove that R has characteristic 2 .
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    Quote Originally Posted by petter View Post
    Suppose R is a finite ring with an odd number of invertible elements. Prove that R has characteristic 2 .
    I denote by R^* the set of invertible elements of R.
    Assume that, for every x\in R^*, x\neq -x. Then, if we associate each element of R^* with its opposite (which is in R^* as well), we partition R^* in pairs, contradicting the fact that the cardinality of R^* is odd.
    As a consequence, there exists x\in R^* such that x=-x. Multiplying by x^{-1}, this implies 1=-1 (where 1 is the unit element of R), or 2=0 (where 2=1+1\in R): the characteristic is 2.
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