Characteristic times Ring Element is Ring Zero

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

Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$.

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

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

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

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

Then: