Characteristic of a Ring

The characteristic of a ring $R$, often denoted $\mathrm{char}(R)$, is defined to be the smallest positive number of copies of the ring's multiplicative identity $1$ that will sum to the additive identity $0$. If such a number does not exist, then the characteristic would be $0$

If a nontrivial ring $R$ does not have any nontrivial zero divisors, then its characteristic is either $0$ or prime. In particular, this applies to all fields, and all Division Rings