TITLE: When are infinite dimensional path algebras hereditary
QUESTION [9 upvotes]: The title says mostly everything.
Suppose we  have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary? Especially in the case that the path algebra is infinite dimensional.
I would appreciate any reference.

REPLY [4 votes]: I assume that you want to unital algebras, so the quivers have only finitely many points. Now let kQ be the path algebra and I an ideal contained in J^2.
Then kQ/I being hereditary implies I=0, see Lemma 4.2.1 in the representation theory book of Benson.
Now kQ is always hereditary, that is mentioned in theorem 4.1.4 in that book and there it quotes a paper by Bergman: Modules over coproducts of rings.
So the general answer might be the same as in the finite dimensional case: kQ/I is hereditary (for an admissible ideal I) for a quiver with finitely many points iff I=0