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Sum-Essential Graphs of Modules

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Pith's one-line read For modules, a tree-shaped proper sum-essential graph is always a star centered at a simple submodule.

desk verdict Solid, modest module-theory paper: the tree/star classification is correct and the generalization to arbitrary modules is genuinely new; refereeing should focus on tightening a few terse steps, not on correctness. read the letter →

arxiv 1908.05921 v1 pith:ZKYR5Z5V submitted 2019-08-16 math.RA math.CO

Jerzy Matczuk , Ali Majidinya This is my paper · ORCID
classification math.RAmath.CO MSC 05C2505C4016D99
keywords sum-essentialgraphproperessentialsubmoduleuniformdimensiontreetriangle-freesemisimplemodulestronglydisjointsubmodules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies two graphs attached to a module $M$: the sum-essential graph, whose vertices are the nontrivial submodules of $M$ and whose edges join submodules whose sum is an essential submodule (one meeting every nonzero submodule), and the proper sum-essential subgraph induced by nonessential submodules. The main result is a classification: if the proper sum-essential graph is a tree, then it is necessarily a star whose center is a simple submodule, and this happens exactly when any two nonessential submodules whose sum is essential are strongly disjoint and one of them is simple. Along the way the authors show that both graphs are connected with diameter at most 3, that finiteness of degrees forces finiteness of the graph, and that the full sum-essential graph is complete or regular only in very restricted forms. The paper thereby translates module-theoretic data, such as uniform dimension and the structure of the socle, into recognizable graph shapes and reads module structure back off graph properties.

What carries the argument

The load-bearing mechanism is the proper sum-essential graph $\mathcal{P}_R(M)$ together with the relation of strong disjointness: two submodules are strongly disjoint when they contain no nonzero isomorphic submodules. Lemma 3.10 recasts this as the condition that corresponding nonzero elements have different annihilators, and as a decomposition property for submodules of their sum. In Theorem 3.11 this relation filters out triangles: with uniform dimension 2, any essential-sum pair of nonessential submodules must be strongly disjoint or a triangle appears. Theorem 3.12 then shows that a tree forces one member of every essential-sum pair to be simple, which collapses the graph to a star centered at that simple submodule.

What would settle it

Compute the adjacency structure of $\mathcal{P}_\mathbb{Z}(M)$ for $M=\mathbb{Z}_{p^2}\oplus\mathbb{Z}_{q^2}$ with distinct primes $p,q$; the paper predicts a 4-cycle and hence not a tree, so if this computation instead produced a tree, Theorem 3.12 would be refuted. Alternatively, a single example of a tree-shaped $\mathcal{P}_R(M)$ without a simple submodule at its center would refute the direction that identifies the tree's center as simple.

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Extended reading notes

Core claim

The central discovery, on the paper's own terms, is Theorem 3.12: for a module $M$ with nonempty proper sum-essential graph $\mathcal{P}_R(M)$, the graph is a tree if and only if every pair of nonessential submodules $A,B$ with $A+B$ essential is strongly disjoint and at least one of $A,B$ is simple; equivalently, $\mathcal{P}_R(M)$ is a star graph whose center is a simple submodule. The proof builds on the triangle-free characterization of Theorem 3.11, which forces uniform dimension 2 and strong disjointness for essential-sum pairs, and then rules out configurations that would create a 4-cycle unless one summand is simple. The resulting module structure is an extension of a simple submodule by a semisimple complement with no isomorphic copies of that simple elsewhere; the paper's examples include $\mathbb{Z}$-modules such as $\mathbb{Z}\oplus \mathbb{Z}_q$ and $\mathbb{Z}_{p^n}\oplus\mathbb{Z}_q$ for distinct primes $p,q$.

Load-bearing premise

The classification leans on the standard module-theoretic facts that every nonzero submodule of a uniform module is essential in it and that a proper submodule of a semisimple module is never essential; if those facts fail, the adjacency tests carrying the proof would stop working.

Editorial extensions

If this is right

  • If the proper sum-essential graph is a tree, the module has uniform dimension 2, so every nonessential submodule is uniform and the graph is a star with a simple center.
  • For any module with a nonempty proper sum-essential graph, the girth is 3, 4, or infinity; if the uniform dimension exceeds 2, the girth is 3.
  • If every vertex of the proper sum-essential graph has finite degree, the graph itself is finite; for the full sum-essential graph, the same condition forces the module to have only finitely many submodules.
  • The sum-essential graph is complete exactly when the module is uniform, or all nonessential submodules are simple with a socle that is a direct sum of two simples; for semisimple modules this reduces to a direct sum of two simple modules.
  • A $k$-regular sum-essential graph is necessarily complete, so regularity does not create genuinely new graph shapes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equivalence in Theorem 3.12 suggests a recipe for building modules with a prescribed star-shaped proper sum-essential graph: start with a simple submodule $S$ with no isomorphic copies elsewhere, add a semisimple complement, and check that every essential-sum pair is strongly disjoint; the paper does not present this as a construction, but it follows directly from the proof.
  • Because strong disjointness has an annihilator reformulation, the tree condition could be checked computationally for finitely generated modules once annihilators of elements are known; this is a test the paper does not carry out.
  • The result implies that a path with three or more vertices cannot occur as a proper sum-essential graph, which is a useful constraint for anyone trying to realize arbitrary graphs as submodule lattices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper introduces two graphs associated with a left module M over a ring R: the sum-essential graph S_R(M), whose vertices are the nontrivial submodules of M with adjacency when the sum of two vertices is essential, and its induced subgraph P_R(M) on the non-essential submodules. The authors study the interplay between module-theoretic properties of M and graph-theoretic properties of these graphs. The main results include: connectedness with diameter at most 3 (Theorem 1.5); criteria for finiteness of degrees and of the graph (Theorems 2.1 and 2.3); a characterization of vertices of degree 1 in P_R(M) (Theorem 2.13); classification of modules for which S_R(M) or P_R(M) is complete (Theorems 3.2 and Corollary 3.3); the fact that k-regularity forces completeness (Theorem 3.6); classifications of triangle-free and tree graphs (Theorems 3.7, 3.11, and 3.12); and results on girth (Corollaries 3.8 and 3.13). The headline theorem (Theorem 3.12) states that P_R(M) is a tree if and only if any two non-essential submodules whose sum is essential are strongly disjoint and one of them is simple, and equivalently P_R(M) is a star with a simple center.

Significance. The central classification is a genuine contribution to the study of graphs from module structures. Theorem 3.12 elegantly ties the graph-theoretic notion of being a tree to the module-theoretic conditions of strong disjointness and simplicity. The proofs are mostly self-contained and use standard techniques such as uniform dimension, essential extensions, and complements. The paper also provides instructive examples, including Z-modules that illustrate the sharpness of the classifications. If the results are correct, they extend prior work on intersection graphs and essential ideal graphs in a natural way. The characterizations are clean, falsifiable, and likely to be of interest to researchers in module theory and algebraic graph Theory.

minor comments (5)
  1. [Theorem 2.3] In the proof of Theorem 2.3, the case P=0 in the definition of A_P is not handled by the given argument, because a complement to 0 in M is M itself, which is not a vertex of P_R(M). However, since soc(M) is essential by Lemma 2.2, we have A_0 = {0}; adding this observation would complete the proof.
  2. [Theorem 3.12] In the proof of (2) implies (3), the sentence 'The statement (2) implies that one of the simples, say S, does not have proper essential extensions in M' is terse. A short justification (e.g., if both simples had proper essential extensions, those extensions would be non-essential vertices whose sum is essential and neither is simple, contradicting (2)) would improve readability.
  3. [Example 2.16(1)] The notation 'M =R R' is unclear; presumably it means M is R viewed as a left R-module. Please clarify.
  4. [Introduction and throughout] There are several minor typographical issues, such as 'a nd' in the Introduction and inconsistent spacing in 'P R(M)' in a few places. A careful proofreading pass is recommended.
  5. [Theorem 3.7] The sentence 'Implications (2) ⇒ (3) ⇒ (1) are clear' could be slightly expanded for (2) ⇒ (3) to confirm that in both listed cases the graph has exactly two vertices.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tree classification in Theorem 3.12 is derived from definitions and standard module-theoretic facts, with no fitted input or load-bearing self-citation.

full rationale

The paper defines the sum-essential graph from the module structure and proves structural equivalences directly. The central classification in Theorem 3.12 does not assume what it proves: condition (2) is a module-theoretic condition, and the proof builds the star from it via Theorem 3.11 and uniform-dimension arguments. The only terse step is the assertion that (2) yields a simple submodule; this is not circular because, taking any nonessential A and a maximal complement C, one has A∩C=0 and A+C≤e M, and C cannot be essential without intersecting A, so condition (2) applies to A and C and forces one of them to be simple. External references, including [5] and the authors' [8], are used as background or for facts the paper says can be seen directly; no uniqueness theorem or prior result is invoked as the load-bearing premise. No fitted parameters, renamed empirical patterns, or self-citation chains occur. Thus the derivation chain is self-contained and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest only on standard module theory and graph definitions. No numbers are fitted to data, no ad hoc parameters are introduced, and no hidden entities are postulated. The graph definitions themselves are the object of study and are characterized directly by the theorems. The axioms listed are the standard facts invoked most heavily: essential submodules, uniform dimension, semisimple behavior, complements, Zorn's lemma, and Schur's lemma.

assumptions (5)
  • standard math In a semisimple module, a proper submodule is never essential; essential submodules coincide with the whole module.
    Used in Lemma 2.2, Proposition 2.5, and Theorem 3.6 to identify PR(soc(M)) with SR(soc(M)) and to describe adjacency via sums equal to M.
  • standard math Every nonzero submodule of a uniform module is essential in that uniform module.
    Used in Lemma 2.2 and Theorems 3.7 and 3.11 to infer essentiality of sums involving submodules of uniform modules.
  • standard math Any submodule disjoint from a given submodule can be extended to a complement, and complements are unique in the situations where degree-one is assumed.
    Used throughout Section 2, especially Lemmas 2.4, 2.8, 2.9 and Theorem 2.13, to translate degree-one into uniqueness of complements.
  • standard math Zorn's lemma applies to chains of submodules in the proof of a largest degree-one submodule.
    Invoked in Theorem 2.18 to bound chains in the family of degree-one submodules containing a fixed submodule.
  • standard math Schur's lemma and finiteness of Hom sets between simple modules control the number of submodules in a direct sum of simples.
    Underlies Example 1.4 and Theorem 2.1(4)(ii), where finiteness of Hom sets is used to bound the number of submodules.

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Pith. "Pith review of Sum-Essential Graphs of Modules." pith.science (2026). https://pith.science/paper/ZKYR5Z5V

@misc{pith2026190805921,
  author       = {Pith},
  title        = {Pith review of: Sum-Essential Graphs of Modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKYR5Z5V}},
  note         = {Machine review of arXiv:1908.05921}
}
abstract

The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the graph $\mathcal{S}_R(M)$ and its subgraph $\mathcal{P}_R(M)$ induced by vertices which are not essential as submodules of $M$ are investigated. The interplay between module properties of $M$ and properties of those graphs is studied.

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Reference graph

Works this paper leans on

10 extracted references · 9 canonical work pages

  1. [8]

    Matczuk, M

    J. Matczuk, M. Nowakowska, E. R Puczy/suppress lowski, Intersection graphs of modules and rings, J. Algebra Appl. 17(3) (2018) 1850131, DOI: 10.1142/S02194988185013 11

    work page doi:10.1142/s02194988185013 2018

  2. [1]

    Akbari, M

    S. Akbari, M. Habibi, A. Majidinya, R. Manaviyat, The inc lusion ideal graph of rings, Comm. Algebra 43(6) (2015)2457-2465, DOI: 10.1080/00927872.2014.894051

    arXiv 2015

  3. [2]

    Akbari, R

    S. Akbari, R. Nikandish and J. Nikmehr, Some results on th e intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013) 1250200, DOI: 10.1142/S0219498812 502003

    work page doi:10.1142/s0219498812 2013

  4. [3]

    Akbari, H

    S. Akbari, H. A. Tavallaee and S. Khakashi Ghezelahmad, I ntersection graph of submodules of a module, J. Algebra Appl. 11 (2012) 1250019, DOI: 10.1142/S02194988 11005452

    work page doi:10.1142/s02194988 2012

  5. [4]

    Amini, B

    A. Amini, B. Amini, E. Momtahan, M.H. Shirdareh, Haghigh i, On a graph of ideals, Acta Math. Hung. 134 (2012) 369-384, DOI: 10.1007/s10474-011-0121-3

    work page doi:10.1007/s10474-011-0121-3 2012

  6. [5]

    Amjadi, The essential ideal graph of a commutative rin g, Asian-Eur

    J. Amjadi, The essential ideal graph of a commutative rin g, Asian-Eur. J. Math. Vol. 11, No. 2 (2018)1850058. DOI: 10.1142/S1793557118500584

    work page doi:10.1142/s1793557118500584 2018

  7. [6]

    Lam, A First Course in Noncommutative Rings, Spring er-Verlag, New York, 2001

    T.Y. Lam, A First Course in Noncommutative Rings, Spring er-Verlag, New York, 2001

    work page 2001

  8. [7]

    Lam, Lectures on Modules and Rings, Springer-Verla g, New York 1999

    T.Y. Lam, Lectures on Modules and Rings, Springer-Verla g, New York 1999

    work page 1999

Show all 10 references
  1. [9]

    H. J. Wang, Graphs associated to co-maximal ideals of com mutative rings, J. Algebra 320 (2008) 2917- 2933, DOI: 10.1016/j.jalgebra.2008.06.020

    2008 doi

  2. [10]

    H. J. Wang, Co-maximal graph of non-commutative rings, Linear Algebra Appl. 430 (2009) 633-641, DOI: 10.1016/j.laa.2008.08.026. Institute of Mathematics, W arsaw University, Banacha 2, 02 -097 W arsaw, Poland E-mail address : jmatczuk@mimuw.edu.pl Department of Mathematical Sc...

    2009 doi

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