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.

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