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