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We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-divisible right $R$-modules are weak-injective. $(iv)$ Homomorphic images of weak-injective right $R$-modules are weak-injective. $(v)$ Homomorphic images of injective right $R$-modules are weak-injective. $(vi)$ Right $R$-modules of weak dimension $"},"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":"1710.00097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-09-29T21:25:34Z","cross_cats_sorted":[],"title_canon_sha256":"8f27a0c091e2d4955f3731454457582d23db0c8bb616b396b08c595dfb5ba619","abstract_canon_sha256":"7c8c6295bf3dd8b6bd7873513a35b54a131353d472d75c866f44879a6142660e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:57.918604Z","signature_b64":"Yeft5n3pcYSG7sQ4lVe7tj7+tQpn0t0UU+Y8kXaZ0m/NQ1jiyuUORC9UrNQzeLNIZNucQRGPTnNS3n91QJTODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44ee8e1fc48ee1faaa458fa2db30eb03763b54178aaa74241abe8fe2712d4193","last_reissued_at":"2026-05-18T00:33:57.918027Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:57.918027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equivalence of Some Homological Conditions for Ring Epimorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alberto Facchini, Zahra Nazemian","submitted_at":"2017-09-29T21:25:34Z","abstract_excerpt":"Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q = R[S^{-1}] = [S^{-1}] R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-divisible right $R$-modules are weak-injective. $(iv)$ Homomorphic images of weak-injective right $R$-modules are weak-injective. $(v)$ Homomorphic images of injective right $R$-modules are weak-injective. $(vi)$ Right $R$-modules of weak dimension $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00097","kind":"arxiv","version":1},"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":"1710.00097","created_at":"2026-05-18T00:33:57.918104+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.00097v1","created_at":"2026-05-18T00:33:57.918104+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.00097","created_at":"2026-05-18T00:33:57.918104+00:00"},{"alias_kind":"pith_short_12","alias_value":"ITXI4H6ER3Q7","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_16","alias_value":"ITXI4H6ER3Q7VKSF","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_8","alias_value":"ITXI4H6E","created_at":"2026-05-18T12:31:21.493067+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/ITXI4H6ER3Q7VKSFR6RNWMHLAN","json":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN.json","graph_json":"https://pith.science/api/pith-number/ITXI4H6ER3Q7VKSFR6RNWMHLAN/graph.json","events_json":"https://pith.science/api/pith-number/ITXI4H6ER3Q7VKSFR6RNWMHLAN/events.json","paper":"https://pith.science/paper/ITXI4H6E"},"agent_actions":{"view_html":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN","download_json":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN.json","view_paper":"https://pith.science/paper/ITXI4H6E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.00097&json=true","fetch_graph":"https://pith.science/api/pith-number/ITXI4H6ER3Q7VKSFR6RNWMHLAN/graph.json","fetch_events":"https://pith.science/api/pith-number/ITXI4H6ER3Q7VKSFR6RNWMHLAN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN/action/storage_attestation","attest_author":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN/action/author_attestation","sign_citation":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN/action/citation_signature","submit_replication":"https://pith.science/pith/ITXI4H6ER3Q7VKSFR6RNWMHLAN/action/replication_record"}},"created_at":"2026-05-18T00:33:57.918104+00:00","updated_at":"2026-05-18T00:33:57.918104+00:00"}