{"paper":{"title":"An Integrally Closed Reduced Ring with McCoy Localizations That Is Neither McCoy nor Locally a Domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"There exists a reduced and integrally closed commutative ring whose localizations at all maximal ideals are McCoy rings, but the ring itself is neither McCoy nor locally a domain.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Haotian Ma","submitted_at":"2026-04-08T18:04:13Z","abstract_excerpt":"We construct an explicit commutative ring $R$ that is reduced and integrally closed, such that $R_{\\mathfrak p}$ is an integrally closed McCoy ring for every maximal ideal $\\mathfrak p$ of $R$, while $R$ itself is not a McCoy ring and is not locally a domain. This gives an affirmative answer to Problem~9 in \\emph{Open Problems in Commutative Ring Theory}. The construction combines Akiba's Nagata-type example, which already yields an integrally closed reduced ring with integrally closed domain localizations and a finitely generated ideal of zero-divisors with zero annihilator, with an explicit "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct an explicit commutative ring R that is reduced and integrally closed, such that R_p is an integrally closed McCoy ring for every maximal ideal p of R, while R itself is not a McCoy ring and is not locally a domain.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the direct product of Akiba's Nagata-type example and the chosen local integrally closed McCoy ring that is not a domain preserves the local McCoy property at all maximal ideals while retaining the global failure of the McCoy condition.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs a reduced integrally closed ring with McCoy localizations at maximal ideals but which is not McCoy and not locally a domain.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There exists a reduced and integrally closed commutative ring whose localizations at all maximal ideals are McCoy rings, but the ring itself is neither McCoy nor locally a domain.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"65f7bf6d3f88836d1b2811e34950026070a5f771d57c055d5a2ca88822874256"},"source":{"id":"2604.07465","kind":"arxiv","version":2},"verdict":{"id":"4c756381-1be7-4707-8bda-458e10c17421","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:25:03.176929Z","strongest_claim":"We construct an explicit commutative ring R that is reduced and integrally closed, such that R_p is an integrally closed McCoy ring for every maximal ideal p of R, while R itself is not a McCoy ring and is not locally a domain.","one_line_summary":"Constructs a reduced integrally closed ring with McCoy localizations at maximal ideals but which is not McCoy and not locally a domain.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the direct product of Akiba's Nagata-type example and the chosen local integrally closed McCoy ring that is not a domain preserves the local McCoy property at all maximal ideals while retaining the global failure of the McCoy condition.","pith_extraction_headline":"There exists a reduced and integrally closed commutative ring whose localizations at all maximal ideals are McCoy rings, but the ring itself is neither McCoy nor locally a domain."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.07465/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}