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In this paper, we prove the following result: let $R$ be a finite ring and one of the f"},"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":"1305.4315","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-19T01:44:27Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"357d44660ecfbb6f58aa4c614dededba3be6f5b6909e5056a984f7deef8aa57f","abstract_canon_sha256":"21545514f110bfb5fc499cecb7d2910a397f48c47d51aaa4df449811a57fa6d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:24.114226Z","signature_b64":"91/vbwDrs00eZpBjpLjMnfxmr6U1rfuJLGwRD3TNzg/Xtq/1FbKABAUKUTrQKI4/VK0SqRORsOJPGZQxH38vBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8548cc3a1d3d3b0a2daadb1a1d36d945d646dd94572ad57b7809e8bf9621757b","last_reissued_at":"2026-05-18T03:25:24.113542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:24.113542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Application of some combinatorial arrays in coloring of total graph of a commutative ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Ghodratollah Aalipour, Saieed Akbari","submitted_at":"2013-05-19T01:44:27Z","abstract_excerpt":"Let $R$ be a commutative ring with unity and $Z(R)$ and ${\\rm Reg}(R)$ be the set of zero-divisors and non-zero zero-divisors of $R$, respectively. 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