Characteristic of a ring with unity element is the order of the unity element

Question

prove that "The characteristic of a ring with unity element is the order of the unity element regarded as a member of the additive group"

Answer 1

The order of the unity element in the additive group of the ring is the least integer $n$ such that $n\cdot1=0$.

But then for any element $a\in\mathcal R$, we have $n\cdot a=n\cdot(1\cdot a)=(n\cdot 1)\cdot a=0\cdot a=0$.

Thus $n$ is also the characteristic of $\mathcal R$.