Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that:


* $I=I.I$;
* $I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a projective left $R$-module)?



Noetherian ring with a "strange" idempotent ideal
Take any idempotent $e$ in a finite dimensional quiver algebra $KQ/L$ , then the ideal $I=AeA$ will be idempotent but only in rare cases it will be projective.
Here an explicit example:
Take the quiver $Q$ with two vertices 1 and 2 (with corresponding primitive idempotents $e\_1$ and $e\_2$) and an arrow a from 1 to 2 and an arrow b from 2 to 1 and let the relations L be $L=<ab>$. Let $A=kQ/L$, which is the Nakayama algebra with Kupisch series [2,3].
The ideal $I=Ae\_2 A$ will be idempotent but not projective.
In the article "Homological theory of idempotent ideals" by Auslander , Platzeck and Todorov ( <https://www.jstor.org/stable/2154190> ) you can find much more about idempotent ideals in Artin algebras and that they are rarely projective.


