Characteristic - Knowlpedia

Let $R$ be a unital ring with identity $1_R$ . The characteristic of $R$ , denoted $\mathrm{char}(R)$ , is the least positive integer $n$ such that $n\cdot 1_R=0$ , if such an $n$ exists; otherwise $\mathrm{char}(R)=0$ .

Remarks

Characteristic controls the arithmetic inside $R$ : the prime subring is the image of $\mathbb{Z}\to R$ , whose kernel is $n\mathbb{Z}$ when $\operatorname{char}(R)=n$ . When $R$ is a field, this image is its prime field. If $R$ is an integral domain (in particular, a field), then $\mathrm{char}(R)$ is either $0$ or a prime number (see characteristic is zero or prime).

Examples

$\mathrm{char}(\mathbb{Z})=0$ .
For a prime $p$ , $\mathrm{char}(\mathbb{F}_p)=p$ .
$\mathrm{char}(\mathbb{Z}/6\mathbb{Z})=6$ , and $\mathrm{char}(M_n(\mathbb{F}_p))=p$ .