Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

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