## Every nonzero Boolean ring has characteristic 2

Prove that a nonzero Boolean ring has characteristic 2.

Let $R$ be a Boolean ring. Note that $1+1 = (1+1)^2 = 1+1+1+1$, so that $1+1 = 0$. Thus the characteristic of $R$ is at most 2. Since $R$ is nontrivial, we have $1 \neq 0$. Thus the characteristic of $R$ is exactly 2.