The characteristic of an integral domain is zero or prime

Remarks

The proof: suppose that the characteristic is $n = a b$ with $1 < a, b < n$. Then $(a \cdot 1)(b \cdot 1) = n \cdot 1 = 0$, but $a \cdot 1 \neq 0$ and $b \cdot 1 \neq 0$. The characteristic is not $1$, because $1 \neq 0$ in an integral domain. In particular, the characteristic of a field is $0$ or a prime number. The ring $\mathbb{Z}/6\mathbb{Z}$ has characteristic $6$, and it is not an integral domain.