Characteristic of Finite Ring is Non-Zero

Theorem
Let $\struct {R, +, \circ}$ be a finite ring with unity.

Then the characteristic of $R$ is not zero.

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

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