I firstly asked the following question on MathStackExchange, but I did not receive any responses, but a short comment. So, I decided to post it here, hoping to receive answers from experts.

Working with Path Algebras, it does not need sophisticated tools to prove for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is 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, talks and lectures, I am inclined that textbooks and notes intentionally skip this classification, may be due to complexity.

It is puzzling to me why is not this question even addressed! "We know that in the aforementioned setting, intersection of a certain class of ideals (left maximals) of $KQ$ is the arrow ideal. Would not it be nice to have an explicit description of each element of this class, like the description we have for simples, indecomposable projectives and injectives? 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.

Link of my question on MathStackExchange: http://math.stackexchange.com/questions/1198691/why-jacobson-but-not-the-left-right-maximals-individually