I am trying to prove that the module $kQe_i$ of paths starting from a node $i$ in an acyclic finite quiver $Q$ is a projective indecomposable module for $kQ$.
The projectivity is clear since I can decompose $kQ$ as $$ kQ=\bigoplus_{i=1}^nkQe_i $$ and therefore all the $kQe_i$ are projective. About the property of being indecomposable, I get stuck very soon. My first idea was to prove that $e_i$ are primitive idempotents, but I have problems showing that $e_i$ are primitives. Any suggestions on how to continue?
In a finite dimensional algebra, to show that a module is indecomposable is equivalent to the condition that its endomorphism ring is local. Here the endomorphism ring is $End_{kQ}(KQe) \cong eKQe$, which has a basis of all paths starting and ending in e. But since your quiver is acyclic, $eKQe$ has e as a basis and is thus isomorphic to the field K and therefore local.