TITLE: Prove that $Re+Rf=Re\oplus R(f-fe)$
QUESTION [2 upvotes]: Let $e,f$ be idempotent elements of a ring $R$. Prove that $Re+Rf=Re\oplus R(f-fe)$.

This post solves the first part of my question. How should we prove the second part, that $e,f$ are orthogonal?
Help me. Thanks a lot.

REPLY [2 votes]: First of all, it is a good idea to say what you have tried and how far you have come in your attempts. That way it is easir to provide a better answer.
That being said, here are some hints: You need to show two things. First, that $e$ and $f-fe$ spans $Re + Rf$, and secondly that $e$ and $f-fe$ are orthogonal. 
For the first, notice that we certainly have a containment $\supseteq$. The opposite inclusion is also straight-forward.
Secondly, what happens if you multiply $e \cdot (f-fe)$?
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