# Direct Summands of PI Rings as Right Ideals

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.
• Thanks for your answer, but if $R$ does not possess identity what is the strategy? – karparvar Apr 27 '16 at 12:35