Let R be a ring with 1. Show that every principal left ideal is generated by an idempotent iff it is a direct summand of ${}_{R}R$

Question

I think there is a more general statement: A left ideal I of R is a direct summand of ${}_{R}R$ $\Leftrightarrow$ I is generated by an idempotent.

the $ \Rightarrow )$ part I got it.

Now suppose $I = Re$ where $e$ is an idempotent element. I don’t know how to conclude that I is a direct summand.

Any hint would be nice.

Answer 1

Hint: Show that $R=Re \ \oplus \ R(1-e)$