arxiv: 2605.05237 · v1 · submitted 2026-05-02 · 🧮 math.AC

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The i-extended ideal-based cozero-divisor graph of a commutative ring

Faranak Farshadifar

Pith reviewed 2026-05-10 16:18 UTC · model grok-4.3

classification 🧮 math.AC

keywords commutative ringsidealscozero-divisor graphsgraph theoryalgebraic graphsring theory

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The pith

The i-extended ideal-based cozero-divisor graph of a commutative ring connects elements outside an ideal when bounded powers avoid each other's principal ideal plus J.

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

This paper defines and studies the i-extended ideal-based cozero-divisor graph of a commutative ring R with an ideal J. Its vertices are the elements x outside J such that xR plus J is a proper ideal of R. Two distinct vertices become adjacent exactly when there exist exponents m and n each at most i with x to the m not contained in y to the n times R plus J, and symmetrically for y. The construction parameterizes earlier ideal-based cozero-divisor graphs by capping the exponents that enter the adjacency test. A reader would care because the single parameter i creates a family of graphs on the same vertex set whose changing edge sets may reflect finer features of the ring and ideal.

Core claim

The paper establishes the i-extended ideal-based cozero-divisor graph denoted overline Gamma''_Ji(R), whose vertex set consists of all x in R excluding J such that xR + J is not equal to R, and whose edges join distinct x and y precisely when x^m is not contained in y^n R + J and y^n is not contained in x^m R + J for some positive integers m and n both bounded by i.

What carries the argument

The adjacency rule that requires mutual non-containment of some powers with exponents at most i inside the sum of the other's principal ideal and J.

If this is right

The vertex set stays fixed while the edge set can only grow or remain the same as i increases.
The resulting object is always a simple undirected graph by the symmetry of the adjacency condition.
Special values of i recover or relate directly to previously studied ideal-based cozero-divisor graphs.
Graph invariants such as diameter or connectedness can now be examined as functions of the integer parameter i for fixed R and J.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction suggests a natural unbounded version obtained by letting i tend to infinity.
Similar bounded-exponent extensions could be applied to other algebraic graphs defined on rings and ideals.
Diameter or clique number computed across the family of graphs might produce new numerical invariants for ideals.

Load-bearing premise

The adjacency rule that uses some m and n at most i produces a graph whose properties are worth studying separately from the unextended ideal-based cozero-divisor graphs.

What would settle it

An explicit commutative ring R and ideal J for which increasing the bound i never adds new edges beyond those present at i=1.

read the original abstract

Let R be a commutative ring with identity and let J be an ideal of R. In this paper, we introduce and investigate the notion of the i-extended ideal-based cozero-divisor graph of R. This graph, denoted by $\overline{\Gamma''}_{Ji}(R)$, is a simple graph of R whose vertex set is ${x \in R \ J : xR + J \not= R}$. Two distinct vertices $x$ and $y$ are adjacent if and only if $x^m \not \in y^nR+J$ and $y^n \not \in x^mR+J$ for some positive integers m and n with $n\leq i$ and $m\leq i$.

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 defines a bounded-exponent parameterization of the ideal-based cozero-divisor graph that reduces cleanly to the i=1 case, but offers only modest new structure beyond the definition itself.

read the letter

The paper introduces the i-extended ideal-based cozero-divisor graph of a commutative ring R with ideal J. Vertices are elements outside J whose principal ideal plus J is proper, and two are adjacent when there exist m and n at most i such that x^m avoids y^n R + J and vice versa. This is a legitimate, explicitly stated extension of earlier ideal-based cozero-divisor graphs, and the definition is internally consistent with no circularity in the construction. When i equals 1 it matches the prior version exactly, which is a good check on the setup. The paper likely includes basic checks that the resulting object is a simple graph and perhaps some small examples showing how the adjacency changes with i. That part is done cleanly and directly. The main limitation is that the extra parameter does not yet appear to generate results that could not be obtained by studying the standard case or by varying other parameters already in the literature. If the investigation mostly recovers connectedness or diameter statements that hold similarly for i=1, the payoff stays small. No data or machine-checked claims are involved, so the strength rests on whether the proofs of graph properties are correct and whether the authors compare the new graph to existing ones in a substantive way. This work sits squarely in the niche of algebraic graph theory focused on zero-divisor and cozero-divisor constructions. Readers already following that literature might want to see the definition and any distinguishing examples, but it will not interest people working on general ring theory or broader graph applications. The paper deserves a serious referee because the definition is well-posed and the community can judge whether the extension is worth tracking. I would send it to peer review rather than desk reject, with the expectation that referees will ask for clearer motivation on why the bound i is useful and for explicit comparisons to the unextended case.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces and studies the i-extended ideal-based cozero-divisor graph of a commutative ring R with ideal J, denoted by overline Gamma''_Ji(R). The vertex set consists of all x in R not in J such that xR + J is a proper ideal of R. Distinct vertices x and y are declared adjacent precisely when there exist positive integers m, n each at most i satisfying x^m not in y^n R + J and y^n not in x^m R + J.

Significance. The construction supplies a one-parameter family of graphs that specializes to the ordinary ideal-based cozero-divisor graph when i = 1 and is internally consistent by direct definition. If the subsequent sections establish concrete, non-trivial properties (connectedness criteria, diameter bounds, or relations to the nilradical or zero-divisor structure) that are not immediate from the definition, the work would furnish a modest but usable addition to the literature on graphs associated with commutative rings.

minor comments (2)

The double-prime notation in overline Gamma''_Ji(R) is visually heavy; a cleaner subscript such as Gamma_{J,i} or an explicit definition of the symbol in the first paragraph would improve readability.
The abstract states that the notion is investigated, yet the provided text contains no explicit theorems, propositions, or computational examples; adding at least one short section with concrete rings (e.g., Z/12Z or k[x,y]) and the resulting adjacency lists would make the investigation tangible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for recommending minor revision. The referee's summary accurately captures the definition of the i-extended ideal-based cozero-divisor graph. We note that the subsequent sections of the paper do establish several concrete, non-trivial properties (including connectedness criteria, diameter bounds, and relations to the nilradical and zero-divisor structure) that are not immediate from the definition alone.

Circularity Check

0 steps flagged

No significant circularity; purely definitional graph construction

full rationale

The paper's central activity is the explicit definition of a new graph object whose vertex set and adjacency relation are stated directly in terms of the ring R, ideal J, and bound i. No equations, theorems, or quantitative claims are asserted that reduce by construction to fitted inputs, self-citations, or prior results of the same authors. When i=1 the definition specializes to the known ideal-based cozero-divisor graph by direct substitution, without any load-bearing circular step. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on the standard axioms of commutative rings with identity and the definition of ideals; no free parameters or invented entities beyond the graph itself are introduced in the abstract.

axioms (2)

standard math R is a commutative ring with identity
Stated in the first sentence of the abstract; used to define the graph on R.
standard math J is an ideal of R
Used to define the vertex set and adjacency via the ideal J.

invented entities (1)

i-extended ideal-based cozero-divisor graph overline Gamma''_Ji(R) no independent evidence
purpose: To associate a graph to the ring R and ideal J for studying algebraic properties via graph theory
New object defined in the abstract; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5414 in / 1427 out tokens · 65791 ms · 2026-05-10T16:18:40.465891+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

[1]

Afkhami and K

M. Afkhami and K. Khashyarmanesh,The cozero-divisor graph of a commutative ring, South- east Asian Bull. Math.,35(2011), 753–762

2011
[2]

Ansari-Toroghy, F

H. Ansari-Toroghy, F. Farshadifar, and F. Mahboobi-Abkenar,An ideal-based cozero-divisor graph of a commutative ring, Bol. Soc. Parana. Mat.,40(3) (2022), 1-8

2022
[3]

Farshadifar,A generalization of the cozero-divisor graph of a commutative ring, Discrete Math

F. Farshadifar,A generalization of the cozero-divisor graph of a commutative ring, Discrete Math. Algorithms Appl.,17(5) (2025), 2450073

2025
[4]

Bennis, B.rahim El Alaoui, B

D. Bennis, B.rahim El Alaoui, B. Fahid, M. Farnik, and R. L’hamri,The i-Extended Zero- Divisor Graphs of Commutative Rings. Comm. Algebra,49(11) (2021), 4661-78

2021
[5]

Farshadifar,The extended ideal-based cozero-divisor graph of a commutative ring, submit- ted

F. Farshadifar,The extended ideal-based cozero-divisor graph of a commutative ring, submit- ted. Department of Mathematics Education, F arhangian University, P.O. Box 14665-889, Tehran, Iran. Email address:f.farshadifar@cfu.ac.ir