# Thread: Characteristic and size of a Ring

1. ## Characteristic and size of a Ring

I need help with the following problem:

Let $R$ be a ring with $m$ elements. Prove that $char(R)$ divides $m$.

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