TITLE: Given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: 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.