When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ideal.
But, I have never seen or found any description of the left or right maximal ideals of the path algebra for a given quiver, even under the assumptions we had above (finite, connected and acyclic). Expect for some simple examples, which repeatedly appear in texts, or talks and lectures, I am inclined that textbooks and notes intentionally skip this classification, may be due to complexity.
It is quite puzzling to me why neither in textbooks, nor in presentations, this question is even addressed! "We know that in the aforementioned setting, intersection of a certain class of ideals of $KQ$ is the arrow ideal. Would not it be nice to know how the elements of this class look like individually? i.e, could one classify all the maximal (right) ideals of such a path algebra, or in a bit more general setting?"
Any reference which might address this question would be highly appreciated.