Characteristic of a ring (reviewed)

The characteristic of a ring is the least positive integer $n$ for which \[ \underbrace{1+\cdots + 1}_n = 0, \] if such an $n$ exists, and $0$ otherwise. Equivalently, it is the exponent of the additive group of the ring.

The characteristic of a field $k$ is either $0$ or a prime number $p$, depending on whether the prime field of $k$ is isomorphic to $\Q$ or $\F_p$.