{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:MHUISE3SWR7OVTMZBLWOVMJWZN","short_pith_number":"pith:MHUISE3S","canonical_record":{"source":{"id":"1706.07775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-06-23T16:45:32Z","cross_cats_sorted":[],"title_canon_sha256":"a2297b5e7c27d06615c29a313b223ee8cbd39502b59f7bed251c62957e8812f1","abstract_canon_sha256":"f49e00cf3c48dfc5cec66a68b7b4707c44204114f8f68d41bb8c49ca166b7fac"},"schema_version":"1.0"},"canonical_sha256":"61e8891372b47eeacd990aeceab136cb5a6af0b33f59ad85a669ac0a2527f90d","source":{"kind":"arxiv","id":"1706.07775","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.07775","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"arxiv_version","alias_value":"1706.07775v1","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.07775","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"pith_short_12","alias_value":"MHUISE3SWR7O","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MHUISE3SWR7OVTMZ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MHUISE3S","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:MHUISE3SWR7OVTMZBLWOVMJWZN","target":"record","payload":{"canonical_record":{"source":{"id":"1706.07775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-06-23T16:45:32Z","cross_cats_sorted":[],"title_canon_sha256":"a2297b5e7c27d06615c29a313b223ee8cbd39502b59f7bed251c62957e8812f1","abstract_canon_sha256":"f49e00cf3c48dfc5cec66a68b7b4707c44204114f8f68d41bb8c49ca166b7fac"},"schema_version":"1.0"},"canonical_sha256":"61e8891372b47eeacd990aeceab136cb5a6af0b33f59ad85a669ac0a2527f90d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:48.236545Z","signature_b64":"YDQ6uasimYRdBvH5K+l8gokRdwhxsXzF69fnRq3QIT90w2D2dlVxnShaUp2Tz4MEmCLPsSNeFonKOQglq5/mBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61e8891372b47eeacd990aeceab136cb5a6af0b33f59ad85a669ac0a2527f90d","last_reissued_at":"2026-05-18T00:41:48.236003Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:48.236003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.07775","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":"2026-05-18T00:41:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+GMBspk93S6PCzKN25Td7sx+QS3ob1Ya9f+ptDy9q4nJHvrzvkEwl8JdHxc1re0Dov93cCxR3U8K5rd5TTxjBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:15:18.587467Z"},"content_sha256":"5c49b212dcc3e03515fb134274feed9135177c3feb6f52fbee66b5afa28bd3e8","schema_version":"1.0","event_id":"sha256:5c49b212dcc3e03515fb134274feed9135177c3feb6f52fbee66b5afa28bd3e8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:MHUISE3SWR7OVTMZBLWOVMJWZN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Existence criteria and expressions of the (b,c)-inverse in rings and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Julio Benitez, Sanzhang Xu","submitted_at":"2017-06-23T16:45:32Z","abstract_excerpt":"Existence criteria for the $(b,c)$-inverse are given.% in terms of annihilators. We present explicit expressions for the $(b,c)$-inverse by using inner inverses. We answer the question when the $(b,c)$-inverse of $a\\in R$ is an inner inverse of $a$. As applications, we give a unified theory of some well-known results of the $\\{1,3\\}$-inverse, the $\\{1,4\\}$-inverse, the Moore-Penrose inverse, the group inverse and the core inverse."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07775","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":"2026-05-18T00:41:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P3x/nA0rBCFUCPXA1O9TJJljqpGP0kLMKxV/MBCZjb0VoADYJQBWtYdfAtZsrirg+sGy57VvKiu1NXxL+crKAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:15:18.587809Z"},"content_sha256":"d2b0f26f03612ec43653ba84f4f3f392d09f6b281d14c9b3346c49a38be434a7","schema_version":"1.0","event_id":"sha256:d2b0f26f03612ec43653ba84f4f3f392d09f6b281d14c9b3346c49a38be434a7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/bundle.json","state_url":"https://pith.science/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/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":"2026-06-20T23:15:18Z","links":{"resolver":"https://pith.science/pith/MHUISE3SWR7OVTMZBLWOVMJWZN","bundle":"https://pith.science/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/bundle.json","state":"https://pith.science/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MHUISE3SWR7OVTMZBLWOVMJWZN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:MHUISE3SWR7OVTMZBLWOVMJWZN","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":"f49e00cf3c48dfc5cec66a68b7b4707c44204114f8f68d41bb8c49ca166b7fac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-06-23T16:45:32Z","title_canon_sha256":"a2297b5e7c27d06615c29a313b223ee8cbd39502b59f7bed251c62957e8812f1"},"schema_version":"1.0","source":{"id":"1706.07775","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.07775","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"arxiv_version","alias_value":"1706.07775v1","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.07775","created_at":"2026-05-18T00:41:48Z"},{"alias_kind":"pith_short_12","alias_value":"MHUISE3SWR7O","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MHUISE3SWR7OVTMZ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MHUISE3S","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:d2b0f26f03612ec43653ba84f4f3f392d09f6b281d14c9b3346c49a38be434a7","target":"graph","created_at":"2026-05-18T00:41:48Z","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":"Existence criteria for the $(b,c)$-inverse are given.% in terms of annihilators. We present explicit expressions for the $(b,c)$-inverse by using inner inverses. We answer the question when the $(b,c)$-inverse of $a\\in R$ is an inner inverse of $a$. As applications, we give a unified theory of some well-known results of the $\\{1,3\\}$-inverse, the $\\{1,4\\}$-inverse, the Moore-Penrose inverse, the group inverse and the core inverse.","authors_text":"Julio Benitez, Sanzhang Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-06-23T16:45:32Z","title":"Existence criteria and expressions of the (b,c)-inverse in rings and its applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07775","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":"sha256:5c49b212dcc3e03515fb134274feed9135177c3feb6f52fbee66b5afa28bd3e8","target":"record","created_at":"2026-05-18T00:41:48Z","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":"f49e00cf3c48dfc5cec66a68b7b4707c44204114f8f68d41bb8c49ca166b7fac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-06-23T16:45:32Z","title_canon_sha256":"a2297b5e7c27d06615c29a313b223ee8cbd39502b59f7bed251c62957e8812f1"},"schema_version":"1.0","source":{"id":"1706.07775","kind":"arxiv","version":1}},"canonical_sha256":"61e8891372b47eeacd990aeceab136cb5a6af0b33f59ad85a669ac0a2527f90d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61e8891372b47eeacd990aeceab136cb5a6af0b33f59ad85a669ac0a2527f90d","first_computed_at":"2026-05-18T00:41:48.236003Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:48.236003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YDQ6uasimYRdBvH5K+l8gokRdwhxsXzF69fnRq3QIT90w2D2dlVxnShaUp2Tz4MEmCLPsSNeFonKOQglq5/mBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:48.236545Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.07775","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c49b212dcc3e03515fb134274feed9135177c3feb6f52fbee66b5afa28bd3e8","sha256:d2b0f26f03612ec43653ba84f4f3f392d09f6b281d14c9b3346c49a38be434a7"],"state_sha256":"d58dc5845d8e4d62f35b66937db1499662331bf5952e08aa165096964b08e381"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5tcqRjn7aEtQUNh3SM4hODaKTZdPradPUkXZXVIwRWSRmDSTqyi9/f56/QqURc+KgZbcYsnrRhhOP+szoQWPDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T23:15:18.589714Z","bundle_sha256":"65771bec944ddc9319e77971be53249e54404171bb7a0fbfa574d5fdd0d389be"}}