Characteristic of a ring

Definition. $\operatorname{char}(R)=n$ if $n\cdot1_R=0$ and no smaller positive integer works. If no such $n$, $\operatorname{char}(R)=0$.

(a) $\operatorname{char}(\mathbb{Z})=0$ (no positive multiple of $1$ equals $0$).

(b) $\operatorname{char}(\mathbb{Z}/n\mathbb{Z})=n$.

(c) $\operatorname{char}(\mathbb{R})=0$.

(d) $\operatorname{char}(\mathbb{F}_p\times\mathbb{F}_p)=p$.

Conclusion. $\operatorname{char}(\mathbb{Z})=\operatorname{char}(\mathbb{R})=0$; $\operatorname{char}(\mathbb{Z}/n\mathbb{Z})=n$; $\operatorname{char}(\mathbb{F}_p\times\mathbb{F}_p)=p$.