{"paper":{"title":"Betti numbers for cochordal zero-divisor graphs of commutative rings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The layered zero-divisor graph of finite chain rings is cochordal, yielding a refined Betti number formula for its edge ideal.","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.AC","authors_text":"Bilal Ahmad Rather","submitted_at":"2026-05-13T14:52:06Z","abstract_excerpt":"This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\\ell\\ge L$, form some integers $k$ and $\\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then cor"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that C(q,L) is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The layered graph C(q,L) with adjacency rule k + ℓ ≥ L accurately captures the zero-divisor relations in the finite chain ring with residue field of order q and nilpotency index L.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Cochordal zero-divisor graphs of chain rings admit refined Betti formulas yielding 2-linear resolutions for the studied quotient rings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The layered zero-divisor graph of finite chain rings is cochordal, yielding a refined Betti number formula for its edge ideal.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"808fdc7f4a1c9301cb2d69bde7ed67c4b9850c203acf2a53d8227faa74accf22"},"source":{"id":"2605.13622","kind":"arxiv","version":1},"verdict":{"id":"df8c41eb-a1bb-44f1-81cc-e71a8fc4766b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:49:38.066878Z","strongest_claim":"We prove that C(q,L) is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal.","one_line_summary":"Cochordal zero-divisor graphs of chain rings admit refined Betti formulas yielding 2-linear resolutions for the studied quotient rings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The layered graph C(q,L) with adjacency rule k + ℓ ≥ L accurately captures the zero-divisor relations in the finite chain ring with residue field of order q and nilpotency index L.","pith_extraction_headline":"The layered zero-divisor graph of finite chain rings is cochordal, yielding a refined Betti number formula for its edge ideal."},"references":{"count":47,"sample":[{"doi":"","year":2003,"title":"S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a completer-partite graph,J. Algebra270(2003) 169–180,https://doi.org/10.1016/ S0021-8693(03)00370-3","work_id":"9b55676f-84df-487b-995e-3750158def84","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1080/00927870802110888","year":2008,"title":"D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring,Comm. Algebra36 (2008) 3073–3092,https://doi.org/10.1080/00927870802110888","work_id":"8dfb848b-b57b-4ca2-bb29-e38b2cb26cba","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1006/jabr.1998.7840","year":1999,"title":"D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring,J. Algebra217(2) (1999) 434–447,https://doi.org/10.1006/jabr.1998.7840","work_id":"d1bd120e-4ada-415b-a1ee-82a44d15dd50","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.disc.2023.113580","year":2023,"title":"G. Arunkumara, P. J. 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