Characteristic of a ring R divides the number of elements m of a ring > Question: Let $R$ be a ring with $m$ elements. Show that the characteristic of $R$ divides $m$. No mention has been made as to whether the elements are distinct so we're going to assume it is. Let $\text{char}(R)=n$. By hypothesis: $\forall x \in R, n\cdot x=x+\ldots +x=0$. From here, I'd like to use Lagrange theorem but it seems I'm short of a pertinent step required to bridge this. Any hint is appreciated.

The characteristic of $R$ is $n$, so there exists $x\in R$ with additive order $n$. Apply Lagrange's theorem to the additive subgroup of $R$ generated by $x$.