Pith Number

pith:42AVZWB6

pith:2026:42AVZWB6AT6UWKTQ4XJQZPGQ7A

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Betti numbers for cochordal zero-divisor graphs of commutative rings

Bilal Ahmad Rather

The layered zero-divisor graph of finite chain rings is cochordal, yielding a refined Betti number formula for its edge ideal.

arxiv:2605.13622 v1 · 2026-05-13 · math.AC · cs.DM · math.CO

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Claims

C1strongest claim

We prove that C(q,L) is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal.

C2weakest 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.

C3one line summary

Cochordal zero-divisor graphs of chain rings admit refined Betti formulas yielding 2-linear resolutions for the studied quotient rings.

References

47 extracted · 47 resolved · 1 Pith anchors

[1] 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 2003

[2] 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 2008 · doi:10.1080/00927870802110888

[3] 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 1999 · doi:10.1006/jabr.1998.7840

[4] G. Arunkumara, P. J. Cameron, T. Kavaskar and T. Tamizh Chelvam, Induced subgraphs of zero-divisor graphs,Discrete Math.346(7) (2023), Art. No. 113580, https://doi.org/10.1016/j.disc.2023.113580. Bett 2023 · doi:10.1016/j.disc.2023.113580

[5] M. Auslander and D. A. Buchsbaum, Homological dimension in local rings, Trans. Amer. Math. Soc.85(1957) 390–405,https://doi.org/10.1090/ S0002-9947-1957-0086822-7 1957

Receipt and verification

First computed	2026-05-18T02:44:17.863236Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e6815cd83e04fd4b2a70e5d30cbcd0f835805ede35aeb2757b748c9e7f80a838

Aliases

arxiv: 2605.13622 · arxiv_version: 2605.13622v1 · doi: 10.48550/arxiv.2605.13622 · pith_short_12: 42AVZWB6AT6U · pith_short_16: 42AVZWB6AT6UWKTQ · pith_short_8: 42AVZWB6

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/42AVZWB6AT6UWKTQ4XJQZPGQ7A \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: e6815cd83e04fd4b2a70e5d30cbcd0f835805ede35aeb2757b748c9e7f80a838

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bded283c732ca6b6cf4020a48376c7a5d7244e167eb2a1bc53c332dd16d25992",
    "cross_cats_sorted": [
      "cs.DM",
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AC",
    "submitted_at": "2026-05-13T14:52:06Z",
    "title_canon_sha256": "d31ee6ad4fdba43ef8d9b029e65e30da757a3080b0b267a732d604b4da970bbe"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13622",
    "kind": "arxiv",
    "version": 1
  }
}