Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?
Let $A = kQ/I$ where $Q$ is a quiver and $I$ are the relations. Then $A$ is hereditary if and only if $Q$ is finite, connected, and acyclic, and $I = (0)$. This is essentially Theorem 1.7 in Chapter VII of Assem, Simson, Skowronski - Elements of the Representation Theory of Associative Algebras.