## Every Boolean ring is commutative

A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove that every Boolean ring is commutative.

Note first that for all $a \in R$, $-a = (-a)^2 = (-1)^2 a^2 = a^2 = a$. Now if $a,b \in R$, we have $a + b = (a+b)^2$ $= a^2 + ab + ba + b^2$ $= a + ab + ba + b$. Thus $ab + ba = 0$, and we have $ab = -ba$. But then $ab = ba$. Thus $R$ is commutative.