Prime characteristic

Setup. The characteristic map $k\mapsto k\cdot1_R$ is a ring homomorphism from $\mathbb{Z}$ to $R$, with kernel $n\mathbb{Z}$.

Integral domain. $R$ is an integral domain, so $\mathbb{Z}/n\mathbb{Z}\cong\operatorname{Im}$ is an integral domain. This forces $n$ to be prime or $0$ (if $n=ab$ with $a,b>1$, then $\bar a\bar b=\bar0$ in $\mathbb{Z}/n\mathbb{Z}$, giving zero divisors in $R$).

Conclusion. The characteristic of an integral domain is $0$ or prime.