For an integral domain $D$, $na = 0$ where $n \in \mathbb{Z}, n\neq 0$ for some $a \in D - \{0\}$, then $D$ is of finite characteristic.
My attempt:
Let $x \in D$, then
$na \cdot x=0$ since $na=0$ by hypothesis.
$$ \begin{split} na \cdot x &= (a+a+ \cdot\cdot\cdot +a)x \\ &= (ax+ax+ \cdot\cdot\cdot+ax) \\ &= a(x+x+\cdot\cdot\cdot+x) \\ &= a \cdot nx \\ &= 0 \end{split} $$ since $na \cdot x=0$. So now we have $a \cdot nx=0$ and we know $a \neq 0$ $\implies nx=0$ since $D$ is an integral domain which has no zero-divisors.
So we have shown that for $n \in \mathbb{Z}-\{0\}$, $nx=0, \forall x \in D$. So $D$ is of finite characteristic.
Besides some feedback about my proof, I would also be interested in some alternate approaches.