I know that if an ideal $I$ is a direct summand of a ring $R$ then it is an idempotent ideal, i.e. $I^2=I$.
My question concerns the rings all of whose idempotent ideals are direct summands. Is there a well-known characterization for such rings?
Certainly, being a direct summand of $R$, any such ideal is generated by an idempotent element.
Thanks for any suggestion!