{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HSHK3QQ3XTXSSWWAZSA6MFOJAQ","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":"c944b32ab3c52e78bdc43de7c545d8ec3422fb05b83da0d5a5ae77a36315f7e4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-14T12:01:44Z","title_canon_sha256":"342f0b2f18709c48aeaefae1238fdb46ee3c4de3df6759ed0aec6dc26060173d"},"schema_version":"1.0","source":{"id":"1108.2863","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.2863","created_at":"2026-05-18T01:16:54Z"},{"alias_kind":"arxiv_version","alias_value":"1108.2863v2","created_at":"2026-05-18T01:16:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.2863","created_at":"2026-05-18T01:16:54Z"},{"alias_kind":"pith_short_12","alias_value":"HSHK3QQ3XTXS","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HSHK3QQ3XTXSSWWA","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HSHK3QQ3","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:f6e82fe47e24b70330fd800f56391323a79ab6885c9e116b415f251b3bac85bf","target":"graph","created_at":"2026-05-18T01:16:54Z","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 ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a unit element of $R$. It was proved that if $R$ is a commutative ring and $\\fm$ is a maximal ideal of $R$ such that $|R/\\fm|=2$, then $G(R)$ is a complete bipartite graph if and only if $(R, \\fm)$ is a local ring. In this paper we generalize this result by showing that if $R$ is a ring (not necessary commutative), then $G(R)$ is a complete $r$-partite graph if a","authors_text":"E. Estaji, M. R. Khorsandi, S. Akbari","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-14T12:01:44Z","title":"On the Unit Graph of a Noncommutative Ring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.2863","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":"sha256:4c555b1b12869067f60583a9c44de7e1e62b12230807494844e0f038cc1c1306","target":"record","created_at":"2026-05-18T01:16:54Z","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":"c944b32ab3c52e78bdc43de7c545d8ec3422fb05b83da0d5a5ae77a36315f7e4","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-14T12:01:44Z","title_canon_sha256":"342f0b2f18709c48aeaefae1238fdb46ee3c4de3df6759ed0aec6dc26060173d"},"schema_version":"1.0","source":{"id":"1108.2863","kind":"arxiv","version":2}},"canonical_sha256":"3c8eadc21bbcef295ac0cc81e615c90410cb7841210a3ac50c36e10c459c6d1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c8eadc21bbcef295ac0cc81e615c90410cb7841210a3ac50c36e10c459c6d1b","first_computed_at":"2026-05-18T01:16:54.897465Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:54.897465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WIgzOEQuXXsvM5LUtsOq/S1nIr0QuToOTaWZUQwaizgLWX5OItx6DlrIwAxj3z0uKLeQyHUej+61YWfdq/VoBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:54.898063Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.2863","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c555b1b12869067f60583a9c44de7e1e62b12230807494844e0f038cc1c1306","sha256:f6e82fe47e24b70330fd800f56391323a79ab6885c9e116b415f251b3bac85bf"],"state_sha256":"cf3d2bb4a0812f8e4f8b7442b9dc72e3217e152fad7f55005a7c58fdc0a26c3b"}