Minimal injective resolutions and Auslander-Gorenstein property for path algebras

Let R be a ring and Q be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(Q, R) of representations of Q by left R-modules. We also extend our formula to all terms of the minimal injective resolution of RQ. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra RQ is k-Gorenstein if and only if Q = (A) over right arrow (n) and R is a k-Gorenstein ring, where n is the number of vertices of Q.