Proposition: Characteristic of an integral domain must be either $0$ or prime number.
I'm confused by this proposition.
I think the characteristic of an integral domain should be always $0$. Suppose it has characteristic $n$. Then $n * a = 0$ for all a of the integral domain.
since n is not $0$ and, if $c * d = 0$ in integral domain, it means $c=0$ or $d=0$, a should be $0$. Hence $n * a$ is not $0$ when $a$ is nonzero. Therefore, characteristic should be always $0$.
What's wrong with my thought?