Arrow ideal of a path algebra
= Arrow ideal of a path algebra
This two-sided ideal is spanned by paths of positive length; its $m$th power is spanned by paths of length at least $m$. The quotient is a product of copies of the ground field indexed by vertices. For a finite acyclic <quiver>, the ideal is nilpotent and equals the <Jacobson radical>. A loop shows that the equality can fail in general.