Is any direct summand of a PI-ring (polynomial identity ring) necessarily idempotent as a right ideal?
The answer is yes for a special case of PI-rings, namely any direct summand of a commutative ring would be an idempotent ideal.
Thanks for any suggestion!
A right ideal of a ring with identity which is a direct summand is always idempotent: if it is $eR$, then $e=e^2\in (eR)^2$, so $eR\subseteq (eR)^2$, and $(eR)^2\subseteq eR$ trivially.
(Are you really asking about PI rings without identities?)