{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HKR45S4PPR4CEJPEFP2PVIG6LM","short_pith_number":"pith:HKR45S4P","canonical_record":{"source":{"id":"1606.02450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-06-08T08:32:35Z","cross_cats_sorted":[],"title_canon_sha256":"c010a96f7f102b6730c7484d6654da3cf94d5ab0a993efa6af6ce18440a26eff","abstract_canon_sha256":"d21ade340b56b723df8b820faaf322edbed949ca630edbb4f80659bca4d85988"},"schema_version":"1.0"},"canonical_sha256":"3aa3cecb8f7c782225e42bf4faa0de5b3428738ed8e635045d4393c0b260b04b","source":{"kind":"arxiv","id":"1606.02450","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02450","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02450v2","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02450","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"HKR45S4PPR4C","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HKR45S4PPR4CEJPE","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HKR45S4P","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HKR45S4PPR4CEJPEFP2PVIG6LM","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-06-08T08:32:35Z","cross_cats_sorted":[],"title_canon_sha256":"c010a96f7f102b6730c7484d6654da3cf94d5ab0a993efa6af6ce18440a26eff","abstract_canon_sha256":"d21ade340b56b723df8b820faaf322edbed949ca630edbb4f80659bca4d85988"},"schema_version":"1.0"},"canonical_sha256":"3aa3cecb8f7c782225e42bf4faa0de5b3428738ed8e635045d4393c0b260b04b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:34.096873Z","signature_b64":"73DVX2V+j0JO9EsMx2xZKBgeNekIlQMDkds/hwd2tKlop2oGTatmhk1uT5mCbs5OfAT4pNAe+Z7UK6MG0ZX0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aa3cecb8f7c782225e42bf4faa0de5b3428738ed8e635045d4393c0b260b04b","last_reissued_at":"2026-05-18T01:12:34.096443Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:34.096443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02450","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":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UaAaj59CyUu8CEsDbGC5CdUEITORP2d1HmO856uT3Ojataw4x6ccz2RLh7D79xeDR7NLnaSY3ttNec/WU7qbCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:09:13.633721Z"},"content_sha256":"69d6e500d23147ee368dab065002f199a89e04455a97c58d54a8a949a3f9cce5","schema_version":"1.0","event_id":"sha256:69d6e500d23147ee368dab065002f199a89e04455a97c58d54a8a949a3f9cce5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HKR45S4PPR4CEJPEFP2PVIG6LM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Centralizer's applications to the inverse along an element","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Huihui Zhu, Jianlong Chen, Pedro Patricio, Xavier Mary","submitted_at":"2016-06-08T08:32:35Z","abstract_excerpt":"In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02450","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":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X803gWHDM7r2Incj0bXgGu0N+PwqsYK7P6CIu5+PbgIHbrWFnG5f59Dli2V3ZHtIK3jbPvWglUuQC3Iv4NxGBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T05:09:13.634493Z"},"content_sha256":"b20a02b893f900c98731dbe66f7d1cd61c1ea4ec089d7a04170e1ead2886af80","schema_version":"1.0","event_id":"sha256:b20a02b893f900c98731dbe66f7d1cd61c1ea4ec089d7a04170e1ead2886af80"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/bundle.json","state_url":"https://pith.science/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/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":"2026-05-31T05:09:13Z","links":{"resolver":"https://pith.science/pith/HKR45S4PPR4CEJPEFP2PVIG6LM","bundle":"https://pith.science/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/bundle.json","state":"https://pith.science/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HKR45S4PPR4CEJPEFP2PVIG6LM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HKR45S4PPR4CEJPEFP2PVIG6LM","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":"d21ade340b56b723df8b820faaf322edbed949ca630edbb4f80659bca4d85988","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-06-08T08:32:35Z","title_canon_sha256":"c010a96f7f102b6730c7484d6654da3cf94d5ab0a993efa6af6ce18440a26eff"},"schema_version":"1.0","source":{"id":"1606.02450","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02450","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02450v2","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02450","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"HKR45S4PPR4C","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HKR45S4PPR4CEJPE","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HKR45S4P","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:b20a02b893f900c98731dbe66f7d1cd61c1ea4ec089d7a04170e1ead2886af80","target":"graph","created_at":"2026-05-18T01:12:34Z","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":"In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.","authors_text":"Huihui Zhu, Jianlong Chen, Pedro Patricio, Xavier Mary","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-06-08T08:32:35Z","title":"Centralizer's applications to the inverse along an element"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02450","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":"sha256:69d6e500d23147ee368dab065002f199a89e04455a97c58d54a8a949a3f9cce5","target":"record","created_at":"2026-05-18T01:12:34Z","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":"d21ade340b56b723df8b820faaf322edbed949ca630edbb4f80659bca4d85988","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-06-08T08:32:35Z","title_canon_sha256":"c010a96f7f102b6730c7484d6654da3cf94d5ab0a993efa6af6ce18440a26eff"},"schema_version":"1.0","source":{"id":"1606.02450","kind":"arxiv","version":2}},"canonical_sha256":"3aa3cecb8f7c782225e42bf4faa0de5b3428738ed8e635045d4393c0b260b04b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3aa3cecb8f7c782225e42bf4faa0de5b3428738ed8e635045d4393c0b260b04b","first_computed_at":"2026-05-18T01:12:34.096443Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:34.096443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"73DVX2V+j0JO9EsMx2xZKBgeNekIlQMDkds/hwd2tKlop2oGTatmhk1uT5mCbs5OfAT4pNAe+Z7UK6MG0ZX0AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:34.096873Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02450","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69d6e500d23147ee368dab065002f199a89e04455a97c58d54a8a949a3f9cce5","sha256:b20a02b893f900c98731dbe66f7d1cd61c1ea4ec089d7a04170e1ead2886af80"],"state_sha256":"9805aaf26ff7610ec27882d4987400ec7d55aa3bf77e56b8ffce11508b6dafd9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o0FCSZNl3bkRZtr9Q1k7lAfPJyWl63jO6jDANW7kKU3pNK1+tD0oNofOoJos4fxYIqSygzMECU07eBTURiE6BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T05:09:13.638779Z","bundle_sha256":"662ed5819d70327665754a45a4812cafa388fc51798cea72c02d389af030a047"}}