{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:JHTVCK4UUDL5MZKVQ6MM3MCMF7","short_pith_number":"pith:JHTVCK4U","canonical_record":{"source":{"id":"1702.02298","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-08T06:17:50Z","cross_cats_sorted":[],"title_canon_sha256":"6c9bd93bedacddf3b33d16b3deca5928bbbb9ea28b165afbca3e21d7935e7a77","abstract_canon_sha256":"84b1647400f2a7dd012eb52bb66263cc135bcd3d04205e26335a2ad0504c3a78"},"schema_version":"1.0"},"canonical_sha256":"49e7512b94a0d7d665558798cdb04c2fd0faf491511f85f6be186d2cbb6c6c68","source":{"kind":"arxiv","id":"1702.02298","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02298","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02298v1","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02298","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"pith_short_12","alias_value":"JHTVCK4UUDL5","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JHTVCK4UUDL5MZKV","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JHTVCK4U","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:JHTVCK4UUDL5MZKVQ6MM3MCMF7","target":"record","payload":{"canonical_record":{"source":{"id":"1702.02298","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-08T06:17:50Z","cross_cats_sorted":[],"title_canon_sha256":"6c9bd93bedacddf3b33d16b3deca5928bbbb9ea28b165afbca3e21d7935e7a77","abstract_canon_sha256":"84b1647400f2a7dd012eb52bb66263cc135bcd3d04205e26335a2ad0504c3a78"},"schema_version":"1.0"},"canonical_sha256":"49e7512b94a0d7d665558798cdb04c2fd0faf491511f85f6be186d2cbb6c6c68","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:06.283273Z","signature_b64":"E+EwTsBsWa++VaRqZ3Kx79O2r/mfw2JsvSCkDBL9GCmfgBc+gwwnq0Pxm5wrgZ+9Lf3haIwEbXbVnoqzJE4xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49e7512b94a0d7d665558798cdb04c2fd0faf491511f85f6be186d2cbb6c6c68","last_reissued_at":"2026-05-18T00:51:06.282810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:06.282810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.02298","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":"2026-05-18T00:51:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4siP9G4CuZj9hoew+xau815KU//VVTQOQQcijgeZe5+ZwtHmuU3eSrrnEL0snC2j2YvAUDS22+dRweVLgoSzBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:55:07.746370Z"},"content_sha256":"f1c0ce5d1755ef1d88810161f21fdbb11ac151f48aab87878f796c8996afac73","schema_version":"1.0","event_id":"sha256:f1c0ce5d1755ef1d88810161f21fdbb11ac151f48aab87878f796c8996afac73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:JHTVCK4UUDL5MZKVQ6MM3MCMF7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Formal triangular matrix ring with nil clean index $4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dhiren Kumar Basnet, Jayanta Bhattacharyya","submitted_at":"2017-02-08T06:17:50Z","abstract_excerpt":"For an element $a \\in R$, let $\\eta(a)=\\{e\\in R\\mid e^2=e\\mbox{ and }a-e\\in \\mbox{nil}(R)\\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\\sup \\{\\mid \\eta(a)\\mid: a\\in R\\}$. In this article we have characterized formal triangular matrix ring $\\begin{pmatrix}A & M0 & B\\end{pmatrix}$ with nil clean index $4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02298","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":"2026-05-18T00:51:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YtwGZRkHlIs/57Z8/aio7gvosr9aKgC3w3j6opOIopbX7bLUYe3nqeVmWi+qK3cVvRpkG8vPnY7G+sI2eGUKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:55:07.746983Z"},"content_sha256":"3fa048185c8b7649bdc5c7a4af3476494aa9ce127cdeac0e8ae7e39f1309d20e","schema_version":"1.0","event_id":"sha256:3fa048185c8b7649bdc5c7a4af3476494aa9ce127cdeac0e8ae7e39f1309d20e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/bundle.json","state_url":"https://pith.science/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/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":"2026-06-09T16:55:07Z","links":{"resolver":"https://pith.science/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7","bundle":"https://pith.science/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/bundle.json","state":"https://pith.science/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JHTVCK4UUDL5MZKVQ6MM3MCMF7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:JHTVCK4UUDL5MZKVQ6MM3MCMF7","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":"84b1647400f2a7dd012eb52bb66263cc135bcd3d04205e26335a2ad0504c3a78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-08T06:17:50Z","title_canon_sha256":"6c9bd93bedacddf3b33d16b3deca5928bbbb9ea28b165afbca3e21d7935e7a77"},"schema_version":"1.0","source":{"id":"1702.02298","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02298","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02298v1","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02298","created_at":"2026-05-18T00:51:06Z"},{"alias_kind":"pith_short_12","alias_value":"JHTVCK4UUDL5","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JHTVCK4UUDL5MZKV","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JHTVCK4U","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:3fa048185c8b7649bdc5c7a4af3476494aa9ce127cdeac0e8ae7e39f1309d20e","target":"graph","created_at":"2026-05-18T00:51:06Z","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":"For an element $a \\in R$, let $\\eta(a)=\\{e\\in R\\mid e^2=e\\mbox{ and }a-e\\in \\mbox{nil}(R)\\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\\sup \\{\\mid \\eta(a)\\mid: a\\in R\\}$. In this article we have characterized formal triangular matrix ring $\\begin{pmatrix}A & M0 & B\\end{pmatrix}$ with nil clean index $4$.","authors_text":"Dhiren Kumar Basnet, Jayanta Bhattacharyya","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-08T06:17:50Z","title":"Formal triangular matrix ring with nil clean index $4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02298","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:f1c0ce5d1755ef1d88810161f21fdbb11ac151f48aab87878f796c8996afac73","target":"record","created_at":"2026-05-18T00:51:06Z","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":"84b1647400f2a7dd012eb52bb66263cc135bcd3d04205e26335a2ad0504c3a78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-08T06:17:50Z","title_canon_sha256":"6c9bd93bedacddf3b33d16b3deca5928bbbb9ea28b165afbca3e21d7935e7a77"},"schema_version":"1.0","source":{"id":"1702.02298","kind":"arxiv","version":1}},"canonical_sha256":"49e7512b94a0d7d665558798cdb04c2fd0faf491511f85f6be186d2cbb6c6c68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49e7512b94a0d7d665558798cdb04c2fd0faf491511f85f6be186d2cbb6c6c68","first_computed_at":"2026-05-18T00:51:06.282810Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:06.282810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E+EwTsBsWa++VaRqZ3Kx79O2r/mfw2JsvSCkDBL9GCmfgBc+gwwnq0Pxm5wrgZ+9Lf3haIwEbXbVnoqzJE4xCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:06.283273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02298","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1c0ce5d1755ef1d88810161f21fdbb11ac151f48aab87878f796c8996afac73","sha256:3fa048185c8b7649bdc5c7a4af3476494aa9ce127cdeac0e8ae7e39f1309d20e"],"state_sha256":"854fd967523d27bb0e9eeedaf614ced0493ff139733fdd80c69b7aafd2c94ade"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mfdnVe5HM1w3PKBQ6CTlFEW6VWiT398cI+pHkGFIm4JnFoNxuork5RinifB9FfmZeqFsZvDIm1vQ0SnC+YGeDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:55:07.750677Z","bundle_sha256":"df632d66fae660243add8c382b5d6389a4ec9137181eb27e0cb0114e55b1482a"}}