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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.2863","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-14T12:01:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"342f0b2f18709c48aeaefae1238fdb46ee3c4de3df6759ed0aec6dc26060173d","abstract_canon_sha256":"c944b32ab3c52e78bdc43de7c545d8ec3422fb05b83da0d5a5ae77a36315f7e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:54.898063Z","signature_b64":"WIgzOEQuXXsvM5LUtsOq/S1nIr0QuToOTaWZUQwaizgLWX5OItx6DlrIwAxj3z0uKLeQyHUej+61YWfdq/VoBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c8eadc21bbcef295ac0cc81e615c90410cb7841210a3ac50c36e10c459c6d1b","last_reissued_at":"2026-05-18T01:16:54.897465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:54.897465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Unit Graph of a Noncommutative Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"E. 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