{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DX3E6YTYB5FH5SABRFAQJR3LHE","short_pith_number":"pith:DX3E6YTY","schema_version":"1.0","canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","source":{"kind":"arxiv","id":"1107.3050","version":2},"attestation_state":"computed","paper":{"title":"Free Cyclic Submodules and Non-Unimodular Vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Joanne L. Hall, Metod Saniga","submitted_at":"2011-07-15T11:56:15Z","abstract_excerpt":"Given a finite associative ring with unity, $R$, and its two-dimensional left module, $^{2}R$, the following two problems are addressed: 1) the existence of vectors of $^{2}R$ that do not belong to any free cyclic submodule (FCS) generated by a unimodular vector and 2) conditions under which such (non-unimodular) vectors generate FCSs. The main result is that for a non-unimodular vector to generate an FCS of $^{2}R$, $R$ must have at least two maximal right ideals of which at least one is non-principal."},"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":"1107.3050","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-15T11:56:15Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ad1cce1858f164ac3c06984bb240b4f4088a0175530c1c2c876720f563b05ca0","abstract_canon_sha256":"ccc2a0429dc5496f659ef0ed28cd497cb16375eaf27ad69a07f2ab6c2a86a2db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:38.345903Z","signature_b64":"LZqZDV8sg0WyhHJ683SPe4KNOKKBDKzV/cM5yKc1H+vue9fwEbDcgipSKbhlF22sbVYza33sSWWu3aG38W+uBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","last_reissued_at":"2026-05-18T04:15:38.345067Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:38.345067Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Free Cyclic Submodules and Non-Unimodular Vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Joanne L. Hall, Metod Saniga","submitted_at":"2011-07-15T11:56:15Z","abstract_excerpt":"Given a finite associative ring with unity, $R$, and its two-dimensional left module, $^{2}R$, the following two problems are addressed: 1) the existence of vectors of $^{2}R$ that do not belong to any free cyclic submodule (FCS) generated by a unimodular vector and 2) conditions under which such (non-unimodular) vectors generate FCSs. The main result is that for a non-unimodular vector to generate an FCS of $^{2}R$, $R$ must have at least two maximal right ideals of which at least one is non-principal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3050","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":"1107.3050","created_at":"2026-05-18T04:15:38.345195+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3050v2","created_at":"2026-05-18T04:15:38.345195+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3050","created_at":"2026-05-18T04:15:38.345195+00:00"},{"alias_kind":"pith_short_12","alias_value":"DX3E6YTYB5FH","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DX3E6YTYB5FH5SAB","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DX3E6YTY","created_at":"2026-05-18T12:26:26.731475+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/DX3E6YTYB5FH5SABRFAQJR3LHE","json":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE.json","graph_json":"https://pith.science/api/pith-number/DX3E6YTYB5FH5SABRFAQJR3LHE/graph.json","events_json":"https://pith.science/api/pith-number/DX3E6YTYB5FH5SABRFAQJR3LHE/events.json","paper":"https://pith.science/paper/DX3E6YTY"},"agent_actions":{"view_html":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE","download_json":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE.json","view_paper":"https://pith.science/paper/DX3E6YTY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3050&json=true","fetch_graph":"https://pith.science/api/pith-number/DX3E6YTYB5FH5SABRFAQJR3LHE/graph.json","fetch_events":"https://pith.science/api/pith-number/DX3E6YTYB5FH5SABRFAQJR3LHE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/action/storage_attestation","attest_author":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/action/author_attestation","sign_citation":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/action/citation_signature","submit_replication":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/action/replication_record"}},"created_at":"2026-05-18T04:15:38.345195+00:00","updated_at":"2026-05-18T04:15:38.345195+00:00"}