Characteristic of a Ring

Definition

The characteristic of a ring $R$ is defined as the smallest positive natural $n\in {\mathbb{N}}^{+}$ such that:

$$\Sigma i:[\mathrm{0..}n).1=0$$

If no such $n$ exists then we say that the characteristic is $0$. Note that this definition relies on LEM due to Limited Principle of Omniscience.