Characteristic of a ring

Definition

Let $R$ be a ring. For a positive integer $n$, write $n \cdot 1 = 1 + 1 + \dots + 1$, with $n$ terms. If there is a positive integer $n$ with $n \cdot 1 = 0$ in $R$, then the characteristic of $R$ is the least such $n$. If there is no such $n$, then the characteristic of $R$ is $0$.

Remarks

The rings $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ have characteristic $0$. For $n \ge 1$, the ring $\mathbb{Z}/n\mathbb{Z}$ has characteristic $n$. The characteristic of $R$ is the nonnegative generator of the kernel of the unique ring homomorphism $\mathbb{Z} \to R$. The characteristic of an integral domain is zero or prime.

Explanations

L1 human

Start with $1$ and keep adding $1$. In the integers you never come back to $0$, so the characteristic is $0$. Modulo $5$, you get $0$ after five steps, so the characteristic is $5$.