**Show that a Boolean ring is commutative**

By definition, a ring R is

*Boolean*if x^2 = x, \forall x \in R.

**Proof.**We need to show that xy = yx for all x,y \in R. So first, we have:

\begin{align*} (x + y)^2 = (x + y) &\iff x^2 + xy + yx + y^2 = x + y \\ &\iff x + xy + yx + y = x + y \\ &\iff xy + yx = 0 \end{align*}

\begin{align*} (y + y) &= (y + y)^2 \\ &= y^2 + 2y + y^2 \\ &= y + y + y + y \\ &= 0 \end{align*}