 Characteristic of a rng

The characteristic of a rng is the least positive integer annihilating every element of its additive abelian group, or zero if no such integer exists. In a unital ring this equals the characteristic of a ring: $n1=0$ implies $nr=(n1)r=0$, and annihilating every element in particular annihilates $1$. In a finite rng it is the exponent of a finite group of the additive group.