Let $R$ be a ring with identity. Prove that $R$ is a Boolean ring if and only if $(a+b)ab=0$ for all $a,b\in R$.

Question

This is my first time to ask question.

Let $R$ be a ring with identity. Prove that $R$ is a Boolean ring if and only if $(a+b)ab=0$ for all $a,b\in R$.

I only proved one implication. Here it is.

Proof: Let $R$ be a ring with identity and let $a,b\in R$. Suppose $R$ is a Boolean ring. Then $a^2=a$ and $b^2=b$, and $R$ is commutative. Now

$$(a+b)ab=a(ab)+b(ab)=\\ (a^2)b+bab=ab+a(b^2)=\\ ab+ab=2ab=0$$ since the characteristic of a Boolean ring is $2$.

Answer 1

The first part is correct. (One minor quibble: the characteristic of the zero ring is $1$.)

The reverse implication can be proved by taking $b=-1$.