A Boolean ring in which if $2a=0$ then $a=0$

Question

In every Boolean ring we have $a^2=a$ for every $a$ in the ring. In some Boolean rings, if $2a=0$ then $a=0$. How to show this ring has just one member? Thanks in advance

Answer 1

Hint: Prove that $(a+a)^2=a+a$ implies $a+a=0$

Answer 2

In a Boolean ring $2a=0$ for every $a$, so "for every $a$, if $2a=0$ then $a=0$" just means "for every $a$ one has $a=0$", which is what you wanted to prove.

To show that $2a=0$ always, do as egreg suggested: polarise $2a=(a+a)^2-2a^2=a+a-2a=0$.