= Finite ring characteristic divides cardinality
{title2=$\operatorname{char}R\mid |R|$}
In a finite unital <ring> $R$, some positive multiple of $1$ is zero, because two of the finitely many successive multiples coincide. If $n=\operatorname{char}R$, the <permutation> $r\mapsto r+1$ has every <cycle of a permutation> of length exactly $n$: returning to $r$ after $k$ steps means $k1=0$, whose least positive solution is $n$. These disjoint cycles partition the <finite set> $R$, so $n$ divides its <cardinality>. The <zero ring> has cardinality and characteristic both one.
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