Abstract
ABSTRACTLet 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 and R is a k-Gorenstein ring, where n is the number of vertices of 𝒬.
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