{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:Z7IWEAFHF3GB4SDELYBWBFWB7R","short_pith_number":"pith:Z7IWEAFH","schema_version":"1.0","canonical_sha256":"cfd16200a72ecc1e48645e036096c1fc487564197d0727b423c64a7121c86adf","source":{"kind":"arxiv","id":"1405.4552","version":1},"attestation_state":"computed","paper":{"title":"Left localizable rings and their characterizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RA","authors_text":"V. V. Bavula","submitted_at":"2014-05-18T20:58:34Z","abstract_excerpt":"A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A c"},"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":"1405.4552","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-05-18T20:58:34Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"11dc38954dbad5ba3a979dbd225a2b91f53f3afd3e9dbb0b201480b7599bbb81","abstract_canon_sha256":"f5656973b3ef8ba3db30d1d9da9283dfc77ff69a9b4e0e4e4125c24a93dd959a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:35.208229Z","signature_b64":"AN3rrdRFOZ9nJ5LIsmuR8Bj6ZQv8f3x/u8cwg5F+w/6Ucc/eI2J13t7AOCVFUJxNQ9OGNUPphUy05B+fZYuLBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfd16200a72ecc1e48645e036096c1fc487564197d0727b423c64a7121c86adf","last_reissued_at":"2026-05-18T02:51:35.207785Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:35.207785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Left localizable rings and their characterizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RA","authors_text":"V. V. Bavula","submitted_at":"2014-05-18T20:58:34Z","abstract_excerpt":"A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4552","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":"1405.4552","created_at":"2026-05-18T02:51:35.207843+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.4552v1","created_at":"2026-05-18T02:51:35.207843+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.4552","created_at":"2026-05-18T02:51:35.207843+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z7IWEAFHF3GB","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z7IWEAFHF3GB4SDE","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z7IWEAFH","created_at":"2026-05-18T12:28:59.999130+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/Z7IWEAFHF3GB4SDELYBWBFWB7R","json":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R.json","graph_json":"https://pith.science/api/pith-number/Z7IWEAFHF3GB4SDELYBWBFWB7R/graph.json","events_json":"https://pith.science/api/pith-number/Z7IWEAFHF3GB4SDELYBWBFWB7R/events.json","paper":"https://pith.science/paper/Z7IWEAFH"},"agent_actions":{"view_html":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R","download_json":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R.json","view_paper":"https://pith.science/paper/Z7IWEAFH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.4552&json=true","fetch_graph":"https://pith.science/api/pith-number/Z7IWEAFHF3GB4SDELYBWBFWB7R/graph.json","fetch_events":"https://pith.science/api/pith-number/Z7IWEAFHF3GB4SDELYBWBFWB7R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R/action/storage_attestation","attest_author":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R/action/author_attestation","sign_citation":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R/action/citation_signature","submit_replication":"https://pith.science/pith/Z7IWEAFHF3GB4SDELYBWBFWB7R/action/replication_record"}},"created_at":"2026-05-18T02:51:35.207843+00:00","updated_at":"2026-05-18T02:51:35.207843+00:00"}