The characteristic of an integral domain is zero or prime

Statement

Let $R$ be an integral domain. Then the characteristic of $R$ is $0$ or a prime number.

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.

Explanations

L1 human

If $6 \cdot 1 = 0$ in a ring, then $(2 \cdot 1)(3 \cdot 1) = 0$. In an integral domain, this forces $2 \cdot 1 = 0$ or $3 \cdot 1 = 0$. So the least $n$ with $n \cdot 1 = 0$ cannot be a product of two smaller numbers.