{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:52O5JWNU3CFZ3SN5PJUCRVMH7O","short_pith_number":"pith:52O5JWNU","canonical_record":{"source":{"id":"1510.03397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-12T19:18:24Z","cross_cats_sorted":[],"title_canon_sha256":"4b21bd6566e4da4ef19677c1360e7fad6846a461a64a9d0d0acb4dfba08535bb","abstract_canon_sha256":"008f8361c5265a2c52987b87e1feb24c6017582bfac021cb8b08205e22d0fd43"},"schema_version":"1.0"},"canonical_sha256":"ee9dd4d9b4d88b9dc9bd7a6828d587fb9c22b86caec611f2762a3fdb0ad4588f","source":{"kind":"arxiv","id":"1510.03397","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03397","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03397v1","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03397","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"pith_short_12","alias_value":"52O5JWNU3CFZ","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"52O5JWNU3CFZ3SN5","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"52O5JWNU","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:52O5JWNU3CFZ3SN5PJUCRVMH7O","target":"record","payload":{"canonical_record":{"source":{"id":"1510.03397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-12T19:18:24Z","cross_cats_sorted":[],"title_canon_sha256":"4b21bd6566e4da4ef19677c1360e7fad6846a461a64a9d0d0acb4dfba08535bb","abstract_canon_sha256":"008f8361c5265a2c52987b87e1feb24c6017582bfac021cb8b08205e22d0fd43"},"schema_version":"1.0"},"canonical_sha256":"ee9dd4d9b4d88b9dc9bd7a6828d587fb9c22b86caec611f2762a3fdb0ad4588f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:26.377463Z","signature_b64":"ggmr5MXcXyeQfTGdw3ZImhjCxKkDYBRHVTDLvEoEAG9vBAupn/uLSePkk8LkCL8xgB1CvAcSulqTE3YQY+RcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee9dd4d9b4d88b9dc9bd7a6828d587fb9c22b86caec611f2762a3fdb0ad4588f","last_reissued_at":"2026-05-18T01:30:26.376785Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:26.376785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.03397","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:30:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sTW8UhdBjer1xyN3SByPuIfXnAokJBAsU1zdI4iZR8d4cCbyXk5Yful4pd6eNxmQb+cyBdg/mNpradhBvo5XDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T20:39:19.775446Z"},"content_sha256":"0ec1b3174fd53c5589d91e1cf22b193b7d00af6a5cfdc63413fe845984bc7298","schema_version":"1.0","event_id":"sha256:0ec1b3174fd53c5589d91e1cf22b193b7d00af6a5cfdc63413fe845984bc7298"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:52O5JWNU3CFZ3SN5PJUCRVMH7O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Projective modules and Gr\\\"obner bases for skew PBW extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Claudia Gallego, Oswaldo Lezama","submitted_at":"2015-10-12T19:18:24Z","abstract_excerpt":"Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincar\\'e-Birkhoff-Witt) extensions. In the present paper we study two aspects of these non-commutative rings: its finitely generated projective modules from a matrix-constructive approach, and the construction of the Gr\\\"obner theory for its left ideals and modules. These two topics could be interesting in future eventual applications of skew $PBW$ extensions in functional linear systems and in non-commutative algebraic geometry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03397","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:30:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qP5bacsXW0tfwPo6BT1bba5HVX99thV6YxocWVvRw3B9lT8k6bx6CF5CCgRonfAxXSC7YdYJOL63PgiZng9oDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T20:39:19.775811Z"},"content_sha256":"d18a650bb1b0d069e7e064ace7ab887f84d3ddd34af8ece0e136e49babcc1ca9","schema_version":"1.0","event_id":"sha256:d18a650bb1b0d069e7e064ace7ab887f84d3ddd34af8ece0e136e49babcc1ca9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/bundle.json","state_url":"https://pith.science/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/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":"2026-06-28T20:39:19Z","links":{"resolver":"https://pith.science/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O","bundle":"https://pith.science/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/bundle.json","state":"https://pith.science/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/52O5JWNU3CFZ3SN5PJUCRVMH7O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:52O5JWNU3CFZ3SN5PJUCRVMH7O","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":"008f8361c5265a2c52987b87e1feb24c6017582bfac021cb8b08205e22d0fd43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-12T19:18:24Z","title_canon_sha256":"4b21bd6566e4da4ef19677c1360e7fad6846a461a64a9d0d0acb4dfba08535bb"},"schema_version":"1.0","source":{"id":"1510.03397","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03397","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03397v1","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03397","created_at":"2026-05-18T01:30:26Z"},{"alias_kind":"pith_short_12","alias_value":"52O5JWNU3CFZ","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"52O5JWNU3CFZ3SN5","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"52O5JWNU","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:d18a650bb1b0d069e7e064ace7ab887f84d3ddd34af8ece0e136e49babcc1ca9","target":"graph","created_at":"2026-05-18T01:30:26Z","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":"Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincar\\'e-Birkhoff-Witt) extensions. In the present paper we study two aspects of these non-commutative rings: its finitely generated projective modules from a matrix-constructive approach, and the construction of the Gr\\\"obner theory for its left ideals and modules. These two topics could be interesting in future eventual applications of skew $PBW$ extensions in functional linear systems and in non-commutative algebraic geometry.","authors_text":"Claudia Gallego, Oswaldo Lezama","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-12T19:18:24Z","title":"Projective modules and Gr\\\"obner bases for skew PBW extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03397","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":"sha256:0ec1b3174fd53c5589d91e1cf22b193b7d00af6a5cfdc63413fe845984bc7298","target":"record","created_at":"2026-05-18T01:30:26Z","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":"008f8361c5265a2c52987b87e1feb24c6017582bfac021cb8b08205e22d0fd43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-12T19:18:24Z","title_canon_sha256":"4b21bd6566e4da4ef19677c1360e7fad6846a461a64a9d0d0acb4dfba08535bb"},"schema_version":"1.0","source":{"id":"1510.03397","kind":"arxiv","version":1}},"canonical_sha256":"ee9dd4d9b4d88b9dc9bd7a6828d587fb9c22b86caec611f2762a3fdb0ad4588f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee9dd4d9b4d88b9dc9bd7a6828d587fb9c22b86caec611f2762a3fdb0ad4588f","first_computed_at":"2026-05-18T01:30:26.376785Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:26.376785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ggmr5MXcXyeQfTGdw3ZImhjCxKkDYBRHVTDLvEoEAG9vBAupn/uLSePkk8LkCL8xgB1CvAcSulqTE3YQY+RcAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:26.377463Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03397","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ec1b3174fd53c5589d91e1cf22b193b7d00af6a5cfdc63413fe845984bc7298","sha256:d18a650bb1b0d069e7e064ace7ab887f84d3ddd34af8ece0e136e49babcc1ca9"],"state_sha256":"f1555ee07898856e5fbc2110215f0bfc8d3028b947ea26a0e66c3bdca7d79fa0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f0KThL6gtHjqQokNvaRo0OjLC41SmnZbrrf7yTtZ1Kv+J83rOX98Gox7U6ZRX0G9c3JtF8SITmwpEnsfyj04Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T20:39:19.777666Z","bundle_sha256":"54acfff53dedf393417f7707c0986f1aaa5b7cb1f15eefc6b7521cd69d7e8e0d"}}