{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:C5N2EAUDQXCXPYLAMY6LFONIL5","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":"0a9ce91ad020bb66d88e06c581b10ea9d212d503400b5c4825ca031598abb6ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-03-12T04:43:35Z","title_canon_sha256":"aa705d95c3def1f1a4b0641e21e02d6ac020ea72a880c79e6c2e3866218afeb3"},"schema_version":"1.0","source":{"id":"1803.04111","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.04111","created_at":"2026-05-18T00:21:32Z"},{"alias_kind":"arxiv_version","alias_value":"1803.04111v1","created_at":"2026-05-18T00:21:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.04111","created_at":"2026-05-18T00:21:32Z"},{"alias_kind":"pith_short_12","alias_value":"C5N2EAUDQXCX","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"C5N2EAUDQXCXPYLA","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"C5N2EAUD","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:93ea492d1730fff0f4f0e7221515d04f671c934191e105c1e31775dc5ce535ec","target":"graph","created_at":"2026-05-18T00:21:32Z","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":"Let $R$ be a finite ring. The commuting probability of $R$, denoted by $\\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. $R$ is called an $n$-centralizer ring if it has $n$ distinct centralizers. In this paper, we compute $\\Pr(R)$ for some $n$-centralizer finite rings.","authors_text":"Dhiren Kumar Basnet, Jutirekha Dutta, Rajat Kanti Nath","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-03-12T04:43:35Z","title":"Commuting probabilities of $n$-centralizer finite rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.04111","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:f10103202f9491614178c49ca062f28c9d828a5456736a00af1647a5f6cb23d4","target":"record","created_at":"2026-05-18T00:21:32Z","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":"0a9ce91ad020bb66d88e06c581b10ea9d212d503400b5c4825ca031598abb6ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-03-12T04:43:35Z","title_canon_sha256":"aa705d95c3def1f1a4b0641e21e02d6ac020ea72a880c79e6c2e3866218afeb3"},"schema_version":"1.0","source":{"id":"1803.04111","kind":"arxiv","version":1}},"canonical_sha256":"175ba2028385c577e160663cb2b9a85f7e1e2e689df98683bae8c2a471fe4b1c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"175ba2028385c577e160663cb2b9a85f7e1e2e689df98683bae8c2a471fe4b1c","first_computed_at":"2026-05-18T00:21:32.606383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:32.606383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gUcOASYdN9h1vpfS3wV8MY85obwzCTv5h+NT+B9IoeOY+4EhEJgZBLhynNf6jIAPOHdsm1IGLeRsnYjQ5ZESDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:32.606893Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.04111","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f10103202f9491614178c49ca062f28c9d828a5456736a00af1647a5f6cb23d4","sha256:93ea492d1730fff0f4f0e7221515d04f671c934191e105c1e31775dc5ce535ec"],"state_sha256":"46de6082b340fe153c3d201b378ea631f6f6aa29bec02c1f582e98ac6d5466bc"}