Characteristic of an integral domain is 0 or prime

Characteristic of an integral domain is 0 or prime: If $D$ is an integral domain, then $\operatorname{char}(D)$ is either $0$ or a prime number $p$ .

Remarks

The proof analyzes the kernel of the canonical map $\mathbb Z\to D$ using the kernel–ideal property, and thereby determines the characteristic of an integral domain. This dichotomy is the starting point for constructing the smallest subfield inside any field (see the corollary on prime subfields).