{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:IGMI6CE5XWSM6S2HSP44JX3BXQ","short_pith_number":"pith:IGMI6CE5","schema_version":"1.0","canonical_sha256":"41988f089dbda4cf4b4793f9c4df61bc1a863fabd4f1c309d2e6fe4a7456d29d","source":{"kind":"arxiv","id":"1505.04526","version":2},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.04526","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-05-18T06:32:16Z","cross_cats_sorted":[],"title_canon_sha256":"9682a86c04e5970ff25b256277f35d3ffec874adef9f238f08ca00c48dc29ee3","abstract_canon_sha256":"cb64f39a91590afa7eb00cc54312e1f4645fe6c722ca0deb711a718390d428f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:26.436616Z","signature_b64":"q9oiFa0FHoRbT2TS6ylU8Run32kR3Mickh0hamE+9ZR846+Gk5tXQTZl2TcaYyOSofswTm5mmpVhx9GU1L6hCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41988f089dbda4cf4b4793f9c4df61bc1a863fabd4f1c309d2e6fe4a7456d29d","last_reissued_at":"2026-05-18T01:16:26.436135Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:26.436135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.04526","created_at":"2026-05-18T01:16:26.436216+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.04526v2","created_at":"2026-05-18T01:16:26.436216+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04526","created_at":"2026-05-18T01:16:26.436216+00:00"},{"alias_kind":"pith_short_12","alias_value":"IGMI6CE5XWSM","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"IGMI6CE5XWSM6S2H","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"IGMI6CE5","created_at":"2026-05-18T12:29:25.134429+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ","json":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ.json","graph_json":"https://pith.science/api/pith-number/IGMI6CE5XWSM6S2HSP44JX3BXQ/graph.json","events_json":"https://pith.science/api/pith-number/IGMI6CE5XWSM6S2HSP44JX3BXQ/events.json","paper":"https://pith.science/paper/IGMI6CE5"},"agent_actions":{"view_html":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ","download_json":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ.json","view_paper":"https://pith.science/paper/IGMI6CE5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.04526&json=true","fetch_graph":"https://pith.science/api/pith-number/IGMI6CE5XWSM6S2HSP44JX3BXQ/graph.json","fetch_events":"https://pith.science/api/pith-number/IGMI6CE5XWSM6S2HSP44JX3BXQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ/action/storage_attestation","attest_author":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ/action/author_attestation","sign_citation":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ/action/citation_signature","submit_replication":"https://pith.science/pith/IGMI6CE5XWSM6S2HSP44JX3BXQ/action/replication_record"}},"created_at":"2026-05-18T01:16:26.436216+00:00","updated_at":"2026-05-18T01:16:26.436216+00:00"}