finite ring with an odd number of invertible elements

**petter** · October 28th 2008, 08:06 AM

Suppose $R$ is a finite ring with an odd number of invertible elements. Prove that $R$ has characteristic $2$ .

**Laurent** · October 28th 2008, 10:36 AM

Originally Posted by petter

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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