# Noetherian ring with a “strange” idempotent ideal

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)?

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 $$L=$$. 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.