{"paper":{"title":"Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Javad Asadollahi, Mohammad Hosein Keshavarz, Rasool Hafezi","submitted_at":"2015-05-18T06:32:16Z","abstract_excerpt":"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, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04526","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}