How to show that every Boolean ring is commutative? A ring $R$ is a Boolean ring provided that $a^2=a$ for every $a \in R$. How can we show that every Boolean ring is commutative?

If $a,b\in R$,
\begin{align}
2ba
&=4ba-2ba\\
&=4(ba)^2-2ba\\
&=(2ba)^2-2ba\\
&=2ba-2ba\\
&=0,
\end{align}
so
\begin{align}
ab
&=ab+0\\
&=ab+2ba\\
&=[ab+ba]+ba\\
&=[(a+b)^2-a^2-b^2]+ba\\
&=[(a+b)-a-b]+ba\\
&=0+ba\\
&=ba.
\end{align}
