# Thread: finite ring with an odd number of invertible elements

1. ## 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 $2$ .

2. 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 $2$.

### prove that the order of a finite ring with 1 is divisible by its characteristic

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