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

Recognition: unknown

Betti numbers for cochordal zero-divisor graphs of commutative rings

Bilal Ahmad Rather

Pith reviewed 2026-05-14 17:49 UTC · model grok-4.3

classification 🧮 math.AC cs.DMmath.CO

keywords zero-divisor graphscochordal graphsBetti numbersedge idealschain ringslinear resolutionscommutative rings

0 comments

The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that a particular layered graph C(q,L), with vertices grouped by valuation layers and edges when the sum of layers meets or exceeds L, is cochordal for finite chain rings. Establishing this property allows the authors to find the type sequence of the graph and to correct an earlier formula for the Betti numbers of the associated edge ideal. These findings are then used for specific families of rings, including quotients of Gaussian integers and truncated polynomial rings over finite fields. The computations indicate that the edge ideals always admit 2-linear resolutions, with Cohen-Macaulay property holding only in complete or degenerate cases. A reader would care because accurate Betti numbers give precise information on the minimal free resolutions and homological dimensions of these algebraic structures.

Core claim

The paper proves that the graph C(q,L) is cochordal. This property allows the determination of its type sequence and the correction of the Betti number formula for the edge ideal of the graph. When specialized to the rings Z_{2^m}[i] and Z_p[x]/(x^c), the computations show that the edge ideals have 2-linear resolutions and the rings are Cohen-Macaulay only in the degenerate or complete-graph cases.

What carries the argument

The layered graph C(q,L) defined by valuation layers with adjacency rule k + ℓ ≥ L, which is shown to be cochordal.

Load-bearing premise

The layered graph C(q,L) with the adjacency rule k + ℓ ≥ L accurately models the zero-divisor relations of the finite chain ring.

What would settle it

Direct computation of the minimal free resolution and Betti numbers for the edge ideal of a small ring such as Z_4[i] or Z_3[x]/(x^2) that differs from the formula derived from the cochordal type sequence.

Figures

Figures reproduced from arXiv: 2605.13622 by Bilal Ahmad Rather.

**Figure 1.** Figure 1: The graph Γ(Z/15Z) = K4,2. The dashed segment marks the false same-class edge that an invalid constructible cover would create [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

**Figure 2.** Figure 2: Arithmetic decision tree for cochordality of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: The graph Γ(Z/8Z), a two-edge star. This small case is useful because it is not complete but is still simple enough to verify by inspection [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

**Figure 4.** Figure 4: Zero divisor graph of Γ(Z/16Z). The example displays a substantial linear strand even for a small ring. The largest nonzero Betti number occurs before the last homological degree, while the projective dimension is 6. n type blocks K corrected Betti sequence pd 8 (2) 1 (2, 1) 2 9 (1, 0) 2 (1) 1 16 (6),(2, 1) 3 (7, 16, 20, 15, 6, 1) 6 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

**Figure 5.** Figure 5: The graph Γ(Z/12Z). The dashed segment between 3 and 9 is not an edge [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Decision rule for the p a q type blocks [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: The actual graph Γ(Z/30Z) is show in [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Zero divisor graph Γ(Z/30Z). n Theorem 7.2 β1 Uncorrected repeated-block value (Theorem 1.10 [15]) Conclusion 30 38 66 Uncorrected value overcounts edges. 42 56 108 Same-class overcount persists. 70 106 182 Larger r amplifies the error [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗

**Figure 9.** Figure 9: Expanded validation path from arithmetic data to homological data. [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗

**Figure 10.** Figure 10: The graph C(3, 4). This definition is modeled on a finite chain ring with residue field of size q and nilpotency index L. In that setting Vk consists of the nonzero elements of valuation k, and the product of elements of valuations k and ℓ is zero precisely when k + ℓ ≥ L. The number of vertices is |V (C(q, L))| = L X−1 k=1 (q − 1)q L−k−1 = q L−1 − 1. This identity will be used repeatedly when projective … view at source ↗

**Figure 11.** Figure 11: The threshold interpretation of C(q, L) [PITH_FULL_IMAGE:figures/full_fig_p044_11.png] view at source ↗

**Figure 12.** Figure 12: The ideal chain of R2 = Z[i]/(4) ∼= Z[i]/(π 4 ) and the sizes |m k | = 24−k [PITH_FULL_IMAGE:figures/full_fig_p049_12.png] view at source ↗

**Figure 13.** Figure 13: The zero-divisor graph Γ(R2) on Z ∗ (R2) = V1 ⊔ V2 ⊔ V3. If A = Z/2 2mZ, then in A every nonzero element can be written uniquely as u2 k with u a unit and 0 ≤ k ≤ 2m − 1. Define the 2–adic valuation w by w(u2 k ) = k and w(0) = +∞. Then the zero divisors of A are the elements with 1 ≤ w(a) ≤ 2m − 1. Consider the sets Wk = {a ∈ A | w(a) = k}, for 1 ≤ k ≤ 2m − 1. Then it is clear that |Wk| = 22m−k−1 , and f… view at source ↗

read the original abstract

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 correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings $\mathbb Z_{2^m}[i]$ and to the truncated polynomial rings $\mathbb Z_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper corrects the Betti formula from the prior Dung-Vu work and gives explicit homological numbers for zero-divisor graphs on two families of finite chain rings.

read the letter

The main takeaway is a correction to the Betti numbers for the edge ideals of cochordal zero-divisor graphs on finite chain rings, together with fresh explicit calculations for the Gaussian rings Z_{2^m}[i] and the truncated polynomial rings Z_p[x]/(x^c). They introduce the layered graph C(q,L) whose vertices sit in valuation layers and whose edges appear exactly when the layer indices sum to at least L. This rule is shown to match the zero-divisor condition via the valuation map on the ring. From there they establish that the graph is cochordal, compute its type sequence, and use the structure to refine the earlier Betti formula. Specializing to the two families yields concrete values for projective dimension, regularity, independence number, height, Hilbert series, and Cohen-Macaulayness, with the result that the resolutions are 2-linear except in the obvious degenerate cases. The stress-test note confirms that the modeling step checks out by direct verification on the valuation and that the counting of minimal generators and syzygies contains no internal inconsistency. That is the solid part: the correction is stated clearly and the new numbers are written down for the chosen rings. The limitation is the narrow focus. Everything stays inside finite chain rings with the specific adjacency rule, so the results do not immediately extend to wider classes of rings or graphs. The paper corrects rather than re-proves the base setup from the 2026 reference, which is reasonable but means the strength of the new formula rests on how well that modeling holds in the checked cases. This is work for people already following zero-divisor graphs and their edge ideals in combinatorial commutative algebra. A reader who needs the corrected formula or the explicit numbers for these two families will find it directly useful. It is worth sending to peer review; the correction and the concrete computations are specific enough that referees can check them without needing a broad new theory.

Referee Report

0 major / 3 minor

Summary. The paper introduces the layered graph C(q,L) whose vertices are partitioned into valuation layers and whose edges satisfy the adjacency rule k + ℓ ≥ L; this graph is shown to model the zero-divisor graph of a finite chain ring with residue field of cardinality q and nilpotency index L. The authors prove that C(q,L) is cochordal, compute its type sequence, correct and refine the Betti-number formula for the edge ideal given in Dung–Vu (Comm. Algebra 2026), and specialize the results to the rings ℤ_{2^m}[i] and ℤ_p[x]/(x^c). They obtain explicit formulas for the projective dimension, Castelnuovo–Mumford regularity, independence number, height, Hilbert series, and Cohen–Macaulay property, concluding that the edge ideals admit 2-linear resolutions and are Cohen–Macaulay only in the complete-graph or degenerate cases.

Significance. If the modeling and homological computations hold, the manuscript supplies the first explicit, parameter-free Betti-number formulas for an infinite family of zero-divisor graphs arising from commutative rings, together with a correction to an earlier formula. The explicit type-sequence description and the verification that the resolutions are 2-linear constitute concrete, reusable data for further work on monomial ideals and graph homological invariants.

minor comments (3)

Abstract, line 3: the phrase “form some integers k and ℓ” is grammatically incomplete; replace with “for integers k, ℓ”.
The specialization sections for ℤ_{2^m}[i] and ℤ_p[x]/(x^c) would benefit from a small table listing the Betti numbers for the first few values of m or c (e.g., m=2,3 and c=2,3) to allow immediate verification of the general formula.
The statement that Cohen–Macaulayness holds “only in the expected degenerate or complete-graph cases” should be accompanied by a precise characterization of those cases in terms of the parameters q and L.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The report correctly identifies the main contributions: the layered graph model C(q,L), the proof that it is cochordal, the correction and refinement of the Betti-number formula from Dung–Vu, and the explicit homological invariants for the specialized chain rings. We will incorporate the minor revisions as recommended.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with an explicit definition of the layered graph C(q,L) whose vertices are partitioned by additive valuation and whose edges are defined by the rule k + ℓ ≥ L; this rule is verified directly from the multiplication in the finite chain ring. Cochordality is established by constructing an explicit ordering, the type sequence is read off from the layers, and the refined Betti numbers are obtained by counting minimal generators and syzygies in each bidegree. The correction to the earlier formula of Dung and Vu is performed on the basis of this independent counting rather than by assuming the prior result. No equation reduces a claimed prediction to a fitted input or to a self-citation chain, and the central claims remain self-contained against the ring-theoretic model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling assumption that the abstract graph C(q,L) faithfully represents the zero-divisor graph of the chain ring; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The graph C(q,L) with vertices arranged by valuation layers and edges when k + ℓ ≥ L models the zero-divisor structure of the finite chain ring.
Explicitly stated as the starting modeling choice in the abstract.

pith-pipeline@v0.9.0 · 5540 in / 1242 out tokens · 38753 ms · 2026-05-14T17:49:38.066878+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

47 extracted references · 37 canonical work pages · 1 internal anchor

[1]

Akbari, H

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

work page doi:10.1080/00927870802110888 2008
[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

work page doi:10.1006/jabr.1998.7840 1999
[4]

Arunkumara, P

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. Betti numbers for cochordal zero-divisor graphs of commutative rings64

work page doi:10.1016/j.disc.2023.113580 2023
[5]

Auslander and D

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
[6]

Banerjee, The regularity of powers of edge ideals,J

A. Banerjee, The regularity of powers of edge ideals,J. Algebraic Combin.41(2) (2015) 303–321,https://doi.org/10.1007/s10801-014-0537-2

work page doi:10.1007/s10801-014-0537-2 2015
[7]

Beck, Coloring of commutative rings,J

I. Beck, Coloring of commutative rings,J. Algebra116(1) (1988) 208–226,https: //doi.org/10.1016/0021-8693(88)90202-5

work page doi:10.1016/0021-8693(88)90202-5 1988
[8]

Bruns and J

W. Bruns and J. Herzog,Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993,https://doi.org/10.1017/ CBO9780511608681

1993
[9]

Algebra324(2010) 3591–3613, https://doi.org/10.1016/j.jalgebra.2010.09.013

R.Chen, Minimal free resolutions of linear edge ideals,J. Algebra324(2010) 3591–3613, https://doi.org/10.1016/j.jalgebra.2010.09.013

work page doi:10.1016/j.jalgebra.2010.09.013 2010
[10]

Chvátal and P

V. Chvátal and P. L. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math.1(1977), 145–162.https://doi.org/10.1016/S0167-5060(08) 70731-3

work page doi:10.1016/s0167-5060(08 1977
[11]

Corso and U

A. Corso and U. Nagel, Specializations of Ferrers ideals,J. Algebraic Combin.28(3) (2008) 425–437,https://doi.org/10.1007/s10801-007-0111-2

work page doi:10.1007/s10801-007-0111-2 2008
[12]

Corso and U

A. Corso and U. Nagel, Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math. Soc.361(3) (2009) 1371–1395,https://doi.org/10.1090/ S0002-9947-08-04636-9

2009
[13]

G. A. Dirac, On rigid circuit graphs,Abh. Math. Semin. Univ. Hambg.25(1961) 71–76, https://doi.org/10.1007/BF02992776

work page doi:10.1007/bf02992776 1961
[14]

Dochtermann, Exposed circuits, linear quotients, and chordal clutters,J

A. Dochtermann, Exposed circuits, linear quotients, and chordal clutters,J. Combin. Theory Ser. A177(2021) 105327,https://doi.org/10.1016/j.jcta.2020.105327

work page doi:10.1016/j.jcta.2020.105327 2021
[16]

Eisenbud,The Geometry of Syzygies, Grad

D. Eisenbud,The Geometry of Syzygies, Grad. Texts in Math. 229, Springer, New York, 2005,https://doi.org/10.1007/b137572

work page doi:10.1007/b137572 2005
[17]

Fernandez-Ramos and P

O. Fernandez-Ramos and P. Gimenez, Regularity3in edge ideals associated to bi- partite graphs,J. Algebraic Combin.39(2014) 919–937,https://doi.org/10.1007/ s10801-013-0473-6

2014
[18]

C. A. Francisco, H. T. Hà and A. Van Tuyl, Splittings of monomial ideals, Proc. Amer. Math. Soc.137(10) (2009) 3271–3282,https://doi.org/10.1090/ S0002-9939-09-09929-8. Betti numbers for cochordal zero-divisor graphs of commutative rings65

2009
[19]

Fröberg, On Stanley–Reisner rings,Banach Center Publ.26(1990) 57–70,https: //doi.org/10.4064/-26-2-57-70

R. Fröberg, On Stanley–Reisner rings,Banach Center Publ.26(1990) 57–70,https: //doi.org/10.4064/-26-2-57-70

work page doi:10.4064/-26-2-57-70 1990
[20]

Fröberg, Betti numbers of fat forests and their Alexander dual,J

R. Fröberg, Betti numbers of fat forests and their Alexander dual,J. Algebraic Combin. 56(2022) 1023–1030,https://doi.org/10.1007/s10801-022-01143-0

work page doi:10.1007/s10801-022-01143-0 2022
[21]

Fröberg, Solution to a conjecture on edge rings with 2-linear resolutions,Comm

R. Fröberg, Solution to a conjecture on edge rings with 2-linear resolutions,Comm. Algebra51(2023) 1447–1450,https://doi.org/10.1080/00927872.2022.2137175

work page doi:10.1080/00927872.2022.2137175 2023
[22]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in alge- braic geometry, Available athttp://www.math.uiuc.edu/Macaulay2/
[23]

H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers,J. Algebraic Combin.27(2) (2008) 215–245,https://doi.org/ 10.1007/s10801-007-0079-y

work page doi:10.1007/s10801-007-0079-y 2008
[24]

Herzog and T

J. Herzog and T. Hibi,Monomial Ideals, Grad. Texts in Math. 260, Springer, London, 2011,https://doi.org/10.1007/978-0-85729-106-6

work page doi:10.1007/978-0-85729-106-6 2011
[25]

https://doi.org/10.1016/j

J. Herzog, T. Hibi and X. Zheng, Dirac’s theorem on chordal graphs and Alexander duality,European J. Combin.25(7) (2004) 949–960,https://doi.org/10.1016/j. ejc.2003.12.008

work page doi:10.1016/j 2004
[26]

Hochster, Cohen–Macaulay rings, combinatorics, and simplicial complexes, inRing Theory II, Lecture Notes in Pure and Applied Mathematics, vol

M. Hochster, Cohen–Macaulay rings, combinatorics, and simplicial complexes, inRing Theory II, Lecture Notes in Pure and Applied Mathematics, vol. 26, Marcel Dekker, New York, 1977, 171–223,https://doi.org/10.1007/978-94-010-1220-1_3

work page doi:10.1007/978-94-010-1220-1_3 1977
[27]

Betti Numbers of Graph Ideals

S. Jacques,Betti Numbers of Graph Ideals, PhD thesis, University of Sheffield, 2004. https://arxiv.org/abs/math/0410107

work page internal anchor Pith review Pith/arXiv arXiv 2004
[28]

Jaramillo and R

D. Jaramillo and R. H. Villarreal, Thev-number of edge ideals,J. Combin. Theory Ser. A177(2021) 105310,https://doi.org/10.1016/j.jcta.2020.105310

work page doi:10.1016/j.jcta.2020.105310 2021
[29]

N. V. R. Mahadev and U. N. Peled, Threshold graphs and related topics,Ann. Discrete Math.56(1995), 1–543.https://doi.org/10.1016/S0167-5060(13)71062-8

work page doi:10.1016/s0167-5060(13)71062-8 1995
[30]

Matsumura,Commutative Ring Theory, Cambridge Stud

H. Matsumura,Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1989,https://doi.org/10.1017/CBO9781139171762

work page doi:10.1017/cbo9781139171762 1989
[31]

Mohammadi and S

F. Mohammadi and S. Moradi, Resolution of unmixed bipartite graphs,Bull. Korean Math. Soc.52(2015) 977–986,https://doi.org/10.4134/BKMS.2015.52.3.977

work page doi:10.4134/bkms.2015.52.3.977 2015
[32]

S. B. Mulay, Cycles and symmetries of zero-divisors,Comm. Algebra30(7) (2002) 3533– 3558,https://doi.org/10.1081/AGB-120004502

work page doi:10.1081/agb-120004502 2002
[33]

H. D. Nguyen and T. Vu, Linearity defect of edge ideals and Fröberg’s theorem,J. Alge- braic Combin.44(1)(2016)165–199,https://doi.org/10.1007/s10801-015-0662-6. Betti numbers for cochordal zero-divisor graphs of commutative rings66

work page doi:10.1007/s10801-015-0662-6 2016
[34]

Pirzada and S

S. Pirzada and S. A. Rather, On the linear strand of edge ideals of some zero divisor graphs,Comm. Algebra51(2) (2023) 620–632,https://doi.org/10.1080/00927872. 2022.2107211

work page doi:10.1080/00927872 2023
[35]

B. A. Rather, Complex Zeros and Log-Concavity in Independent Domination Polyno- mials of Zero Divisor Graphs of Commutative Rings,Theor. Comput. Sci.1058(2025), Art. No. 115594,https://doi.org/10.1016/j.tcs.2025.115594

work page doi:10.1016/j.tcs.2025.115594 2025
[36]

B. A. Rather, Independent domination polynomial of comaximal graphs of com- mutative rings,Algebra Colloq.33(2) (2026) 243–258,https://doi.org/10.1142/ S1005386726000222

2026
[37]

B. A. Rather, Complex zeros of independent domination polynomials of zero divisor graphs,Soft Comput.(2026),https://doi.org/10.1007/s00500-026-11308-9

work page doi:10.1007/s00500-026-11308-9 2026
[38]

B. A. Rather, Betti numbers of edge ideals of some graphs with application to graphs assigned to groups,Filomat38(2024) 2185–2204,https://doi.org/10.2298/ FIL2406185R

2024
[39]

B. A. Rather, Homological invariants of edge ideals of Wollastonite graphs,J. Comb. Optim.(2026),https://doi.org/10.1007/s10878-025-01389-x

work page doi:10.1007/s10878-025-01389-x 2026
[40]

B. A. Rather and J. Wang, Homological invariants of edge ideals of power graphs of finite groups,Filomat40(2) (2026) 739–750,https://doi.org/10.2298/FIL2602739R

work page doi:10.2298/fil2602739r 2026
[41]

B. A. Rather, M. Imran and A. Diene, The linear strand of edge ideals of comaximal graphs of commutative rings,Comm. Algebra52(2024) 1486–1500,https://doi.org/ 10.1080/00927872.2023.2263559

work page doi:10.1080/00927872.2023.2263559 2024
[42]

B. A. Rather, M. Imran and S. Pirzada, Linear strand of edge ideals of zero divisor graphs of the ringZ n,Comm. Algebra52(2024) 5069–5085,https://doi.org/10. 1080/00927872.2024.2363953

work page arXiv 2024
[43]

Singh and R

P. Singh and R. Verma, Betti numbers of edge ideals of some split graphs,Comm. Algebra48(2020) 5026–5037,https://doi.org/10.1080/00927872.2020.1777559

work page doi:10.1080/00927872.2020.1777559 2020
[44]

Spiroff and C

S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors,Comm. Algebra39(7) (2011) 2338–2348,https://doi.org/10.1080/ 00927872.2010.488675

work page arXiv 2011
[45]

R. P. Stanley,Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathe- matics, vol. 41, Birkhäuser, 1996,https://doi.org/10.1007/b139094

work page doi:10.1007/b139094 1996
[46]

R. H. Villarreal,Monomial Algebras, 2nd ed., Monographs and Research Notes in Math- ematics, CRC Press, Boca Raton, 2015,https://doi.org/10.1201/b18224

work page doi:10.1201/b18224 2015
[47]

Wang, Zero-divisor graphs of finite commutative rings,J

H.-J. Wang, Zero-divisor graphs of finite commutative rings,J. Algebra296(2006) 47–58. Betti numbers for cochordal zero-divisor graphs of commutative rings67

2006
[48]

Woodroofe, Matchings, coverings, and Castelnuovo–Mumford regularity,J

R. Woodroofe, Matchings, coverings, and Castelnuovo–Mumford regularity,J. Commut. Algebra6(2) (2014) 287–304,https://doi.org/10.1216/JCA-2014-6-2-287

work page doi:10.1216/jca-2014-6-2-287 2014