TITLE: Characteristic of an integral domain must be either $0$ or a prime number.
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: Hint $\ $ Whenever you have problems understanding such an abstract statement you should look at concrete instances. For example $\,\rm \Bbb Z\ mod\ 3\ $ has characteristic $3$ because $ 3n := n + n + n \equiv 0\ $ for $ n\equiv 0,1,2.\:$ See how you argument breaks down in this simple concrete case, then generalize.
Note that $\,m\cdot a\,$ does not denote an element obtained by applying the ring multiplication to two elements of the ring. Rather, the $m$'th multiple $\,m\cdot a\,$ is the additive analog of the $m$'th power $a^m.$ In the first case we add $m$ copies of $a$ to obtain $\,m\cdot a\,$ and the second we multiply them to get $\,a^m.\,$ They are both well-defined operations in any ring. 
One may rigorously define such operations by recursion, viz. 
$$\begin{eqnarray} 0\cdot a \,&=&\, 0 \\ (1+n)\cdot a\,&=&\, a + n\cdot a\end{eqnarray}$$