Characteristic

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 $\mathrm{char}(R)=0$.

Characteristic controls the arithmetic inside $R$: the prime field (or prime subring) embeds via $\mathbb{Z}\to R$ with kernel $n\mathbb{Z}$. 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 $\mathrm{char}(M_n(\mathbb{F}_p))=p$.