The characteristic of an integral domain is 0 or prime

Let $𝐷$ be an integral domain. Then its characteristic $c h a r (𝐷)$ is either 0 or prime. #m/thm/ring

Proof

Suppose that $1$ has order $𝑛 = 𝑠 𝑡$ where $1 \leq 𝑠, 𝑡 \leq 𝑛$ . Then

0 = 𝑛 \sum 1 = (𝑠 \sum 1) (𝑡 \sum 1)

so either $𝑠 = 𝑛$ or $𝑡 = 𝑛$ . Thus $𝑛$ is prime.

#state/tidy | #lang/en | #SemBr