I'm struggling to resolve an exercise in "Methods of Representation Theory" of Curtis & Reiner.
Let $A$ be a semisimple ring and let $e \in A$ an idempotent different from zero. Show that if $e$ cannot be written as the sum of 2 idempotents $f, f'$ different from zero such that $ff' = f'f = 0$, then the left ideal $Ae$ is simple. I took a left ideal $I$ non-zero, such that $I \subset Ae$ and I want to show that $Ae = I$. Since A is semisimple, there exists an idempotent $f$ such that $I = Af$ and another idempotent $f'$ such that $A = Af \oplus Af'$. Using that, I wrote $e = ef + ef'$ and I proved that $ef$ and $ef'$ are idempotents but I can't prove that $ef' * ef = 0$. Does someone see how to do this ?
Thank you