Given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.

Question

Let $R$ be a ring with identity. An element $a \in R$ it is idempotent if $a^2=a$.

Show that given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.

Remark:

I'm trying the following $a+b = (a+b)^2 = (a+b)(a+b) = a^2+ab+ba+b^2 = a+ab+ba+b \Rightarrow ab = - ba$. I am not able to conclude from this, I tried to $(ab)^2 = (-ba)^2$ but I can not conclude.

Thanks for your help.

Answer 1

I was able to do so using the fact that $ab = -ba$.

We have that

$ab = aabb = -abab = -baba = bbaa = ba $

I believe so is right.