Characteristic times Ring Element is Ring Zero

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with unity, where the unity of $R$ is $1$.

Let the characteristic of $R$ be $n$.

Then:
 * $\forall a \in R: n \cdot a = 0$

Proof
If $a = 0$ then $n \cdot a = 0$ is immediate.

So let $a \in R: a \ne 0$.

Then: