Show that a Boolean ring is a commutative ring.

Question

I have a question about (https://math.stackexchange.com/q/10279)'s proof to this. I also asked as a comment but I am unsure whether it will be replied to since the post was made 8 years ago?

My question to their proof is: How does $x+xy+yx+y=x+y$ imply $xy=yx$? Doesn't $xy=yx$ imply $xy-yx=0$, and so how is $xy+yx=0$?

Answer 1

In a Boolean ring $(-1)^2=-1$, so $1=-1$. In particular $$xy-yx=xy+yx$$