{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:FKB44DNPZOF7FIOA2ZNIQ7VYMF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"086425861a390d4b288b4732522b1b251ddef4c4725442b06e1774cbce947707","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-01-07T22:58:07Z","title_canon_sha256":"54cf5a5285e0f503cbb3c7130b59f627264a1bbdbdfac107414b71444ae6713a"},"schema_version":"1.0","source":{"id":"1801.02260","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.02260","created_at":"2026-05-18T00:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1801.02260v1","created_at":"2026-05-18T00:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02260","created_at":"2026-05-18T00:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"FKB44DNPZOF7","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"FKB44DNPZOF7FIOA","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"FKB44DNP","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:5856660dccb6ab9cd6d04ff77397ae91be50f8202e6ce19764d6c30eaa77c018","target":"graph","created_at":"2026-05-18T00:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"It is well-known that a class of all modules, which are torsion-free with respect to a set of ideals, is closed under injective envelopes. In this paper, we consider a kind of a dual to this statement - are the divisibility classes closed under flat covers? - and argue that this is seldom the case. More precisely, we show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then","authors_text":"Michal Hrbek","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-01-07T22:58:07Z","title":"Divisibility classes are seldom closed under flat covers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02260","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b1f9a7315380658182c3a964c8e1097c1097f5eb692246ef823857230863c685","target":"record","created_at":"2026-05-18T00:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"086425861a390d4b288b4732522b1b251ddef4c4725442b06e1774cbce947707","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-01-07T22:58:07Z","title_canon_sha256":"54cf5a5285e0f503cbb3c7130b59f627264a1bbdbdfac107414b71444ae6713a"},"schema_version":"1.0","source":{"id":"1801.02260","kind":"arxiv","version":1}},"canonical_sha256":"2a83ce0dafcb8bf2a1c0d65a887eb861546e9cbc70aee52897c2fa44d030d75d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a83ce0dafcb8bf2a1c0d65a887eb861546e9cbc70aee52897c2fa44d030d75d","first_computed_at":"2026-05-18T00:26:32.270505Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:32.270505Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EQRvaFiU5oUGKeyfSKXcewVMcOpNfZxRyJVBszCdYeIJQ3PmVrUDs38JY67yNpSZtTFqGaNK0hg0RoRRcSczBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:32.271117Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.02260","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1f9a7315380658182c3a964c8e1097c1097f5eb692246ef823857230863c685","sha256:5856660dccb6ab9cd6d04ff77397ae91be50f8202e6ce19764d6c30eaa77c018"],"state_sha256":"75051894da63cf49b6e133fd06afd942ffd275a165693d82ca1dce913fac5141"}