{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:DX3E6YTYB5FH5SABRFAQJR3LHE","short_pith_number":"pith:DX3E6YTY","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"},"canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","source":{"kind":"arxiv","id":"1107.3050","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3050","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3050v2","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3050","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"pith_short_12","alias_value":"DX3E6YTYB5FH","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DX3E6YTYB5FH5SAB","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DX3E6YTY","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:DX3E6YTYB5FH5SABRFAQJR3LHE","target":"record","payload":{"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"},"canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","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"},"source_kind":"arxiv","source_id":"1107.3050","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:15:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VyYV3MGPcZto/LrMRBzu6av8ou5iBSiy+TOFF3Fev9WGfmxfeyEE5PirqWngsGPA8XoJnwTt57MmyabjQkddCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:53:29.830663Z"},"content_sha256":"5fc9c205f2b455803482f2294a92c6bf267ffffcadfea6fe22886a517e4181af","schema_version":"1.0","event_id":"sha256:5fc9c205f2b455803482f2294a92c6bf267ffffcadfea6fe22886a517e4181af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:DX3E6YTYB5FH5SABRFAQJR3LHE","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:15:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6lGWDnkKHEsIS9h9xkcfOkO6Z829ty5gz1yOOnXvm4mpBzhz0IgTxlanODQJOwDynj3BUgSC1nThAYhpYsHOCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:53:29.831004Z"},"content_sha256":"5cadbc53f9dbb64b322855f267f7e2d0650f3e2b7bb8915c097ea1bbaad1f2a0","schema_version":"1.0","event_id":"sha256:5cadbc53f9dbb64b322855f267f7e2d0650f3e2b7bb8915c097ea1bbaad1f2a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/bundle.json","state_url":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T02:53:29Z","links":{"resolver":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE","bundle":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/bundle.json","state":"https://pith.science/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DX3E6YTYB5FH5SABRFAQJR3LHE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:DX3E6YTYB5FH5SABRFAQJR3LHE","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":"ccc2a0429dc5496f659ef0ed28cd497cb16375eaf27ad69a07f2ab6c2a86a2db","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-15T11:56:15Z","title_canon_sha256":"ad1cce1858f164ac3c06984bb240b4f4088a0175530c1c2c876720f563b05ca0"},"schema_version":"1.0","source":{"id":"1107.3050","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3050","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3050v2","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3050","created_at":"2026-05-18T04:15:38Z"},{"alias_kind":"pith_short_12","alias_value":"DX3E6YTYB5FH","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DX3E6YTYB5FH5SAB","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DX3E6YTY","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:5cadbc53f9dbb64b322855f267f7e2d0650f3e2b7bb8915c097ea1bbaad1f2a0","target":"graph","created_at":"2026-05-18T04:15:38Z","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":"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.","authors_text":"Joanne L. Hall, Metod Saniga","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-15T11:56:15Z","title":"Free Cyclic Submodules and Non-Unimodular Vectors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3050","kind":"arxiv","version":2},"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":"sha256:5fc9c205f2b455803482f2294a92c6bf267ffffcadfea6fe22886a517e4181af","target":"record","created_at":"2026-05-18T04:15:38Z","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":"ccc2a0429dc5496f659ef0ed28cd497cb16375eaf27ad69a07f2ab6c2a86a2db","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-07-15T11:56:15Z","title_canon_sha256":"ad1cce1858f164ac3c06984bb240b4f4088a0175530c1c2c876720f563b05ca0"},"schema_version":"1.0","source":{"id":"1107.3050","kind":"arxiv","version":2}},"canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1df64f62780f4a7ec801894104c76b3900dd1891f176ea1b32817c53ed5b0ff1","first_computed_at":"2026-05-18T04:15:38.345067Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:38.345067Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LZqZDV8sg0WyhHJ683SPe4KNOKKBDKzV/cM5yKc1H+vue9fwEbDcgipSKbhlF22sbVYza33sSWWu3aG38W+uBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:38.345903Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.3050","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5fc9c205f2b455803482f2294a92c6bf267ffffcadfea6fe22886a517e4181af","sha256:5cadbc53f9dbb64b322855f267f7e2d0650f3e2b7bb8915c097ea1bbaad1f2a0"],"state_sha256":"c3846389d987a56e4b7d2d2fb1e2e5ebee26b309e16c9abb62eb4054b35b33b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7gDRXyZMqzDi4GBUyY85xzcmUbVbssb665RxuoXhJ4rkSIq5X94Zd05CEg1r828SxmvC1E1711N0xV7RSGINAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T02:53:29.833070Z","bundle_sha256":"637b9399d5cd602cfbe8f3bd34293aad14c53b790e3f0a09f5faaf69100e49bd"}}