I need help with the following problem:
Let $\displaystyle R $ be a ring with $\displaystyle m$ elements. Prove that $\displaystyle char(R)$ divides $\displaystyle m$.
Thanks in advance.
$\displaystyle (R,+)$ is a group of order $\displaystyle m.$. let $\displaystyle d=\text{char}(R)$ and $\displaystyle m=kd + r,$ where $\displaystyle 0 \leq r < d.$ suppose that $\displaystyle 0 < r < d.$ by Lagrange theorem for finite groups $\displaystyle m \cdot 1_R=0.$ thus $\displaystyle 0=(kd + r) \cdot 1_R = r \cdot 1_R,$
which contradicts the definition of "charateristic" because our assumption was that $\displaystyle 0 < r < d.$ hence $\displaystyle 0 < r < d$ is impossible and so we must have $\displaystyle r=0,$ i.e. $\displaystyle m=kd.$