Characteristic of Finite Ring is Non-Zero

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with unity whose order is finite.

Then the characteristic of $R$ is not zero.

Proof
We have that $\left({R, +, \circ}\right)$ is finite, so its additive group $\left({R, +}\right)$ is likewise finite.

The result follows by Finite Group Elements of Finite Order and the definition of characteristic.