Every nonzero Boolean ring has characteristic 2

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.27
Solution: 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.