Pith. sign in

REVIEW 3 major objections 5 minor 25 references

On Divisor Topology of Modules over Domains

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper defines a divisor topology on the cyclic submodules of a module over a domain and uses it to characterize uniserial, pseudo-simple, and finitely cogenerated modules, claiming compactness exactly when there are finitely many…

desk verdict Central compactness theorem is false; the module-level divisor topology is a sensible construction, but the paper needs major revision before publication. read the letter →

arxiv 2506.01179 v1 pith:7UNAYW4S submitted 2025-06-01 math.AC math.GN

Ünsal Tekir , Uğur Yiğit , Mesut Buğday , Suat Koç This is my paper
classification math.ACmath.GN MSC 13A1554H1016D60
keywords divisortopologymoduleoveradomainpseudo-simpleuniserialfinitelycogeneratedAlexandrovspacecompactnessBaire
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

The paper introduces a topology, the divisor topology $D(M)$, on the set of nonzero nongenerator elements of a module $M$ over a domain, identifying two elements when they generate the same cyclic submodule. The basic open sets $U_m$ collect the classes $[n]$ for which $n$ divides $m$, so open neighborhoods record divisibility by cyclic submodules. The paper's aim is to show that standard topological properties of $D(M)$ are algebraic characterizations: $D(M)$ is nested exactly for uniserial modules, Noetherian exactly for simple modules or vector spaces over a field, and $T_1$/discrete exactly for the newly defined pseudo-simple modules. It also claims that $D(M)$ is compact if and only if $M$ has finitely many simple submodules, which would make compactness a topological certificate for finite socle and finite cogeneration. A sympathetic reader should read the compactness claim as the paper's headline structural result, with the divisor topology as the mechanism connecting the two sides.

What carries the argument

The central object is the divisor topology $D(M)$ on the set $EC(M^{\#})$ of equivalence classes $[m]$ of nonzero nongenerators under the relation $Rm = Rn$, with the basis of open sets $U_m = \{[n] : n \mid m\}$, equivalently $\{[n] : Rm \subseteq Rn\}$. Because $U_m$ is the smallest open set containing $[m]$, $D(M)$ is an Alexandrov space and closure is computed by divisibility: $[n]$ belongs to the closure of $[m]$ exactly when $m \mid n$. This basis turns inclusion of cyclic submodules into inclusion of open sets, so module-theoretic finiteness conditions---simple submodules, uniseriality, vector-space structure---are read off as topological separation, order, and compactness properties. The algebraic notion of pseudo-simple module---every proper cyclic submodule is simple---is introduced as the exact counterpart of the $T_1$/discrete/Hausdorff behavior.

What would settle it

Compute $D(M)$ for $M = \mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})$. The module has exactly one simple submodule, $\{0\} \times (\mathbb{Z}/2\mathbb{Z})$, so the stated criterion predicts compactness. But among the basic open sets, only $U_{(a,0)}$ can contain classes $[(p,0)]$ for primes $p$, and each such set contains only finitely many of them; the cover by all basic open sets therefore has no finite subcover for the infinitely many classes $[(p,0)]$. This makes $D(M)$ noncompact and would refute the claimed equivalence.

Watch

Extended reading notes

Core claim

The central claim is that the divisor topology $D(M)$ is a faithful topological shadow of cyclic-submodule structure in modules over domains. For a nonzero element $m$ that does not generate $M$, the basic open set $U_m$ contains exactly the classes $[n]$ whose cyclic submodule $Rn$ contains $Rm$, so inclusions of cyclic submodules become inclusions of basic open sets. On this basis the paper characterizes: $M$ is uniserial iff $D(M)$ is nested; $M$ is simple or a vector space over the field $R$ iff $D(M)$ is Noetherian; $M$ is pseudo-simple---every proper cyclic submodule is simple---iff $D(M)$ is a $T_1$ space, equivalently a Hausdorff space, equivalently discrete; and $D(M)$ is compact iff the family $\{Rm : m \in M^{\#}\}$ has finitely many minimal elements, iff $M$ has finitely many simple submodules. It further proves that $D(M)$ is always an Alexandrov space, first countable, and Baire for factorial modules, and that compactness implies $M$ is finitely cogenerated.

Load-bearing premise

The compactness characterization assumes that every nonzero cyclic submodule generated by a nonzero nongenerator contains a minimal nonzero cyclic submodule; without this well-foundedness of the family $\{Rm : m \in M^{\#}\}$, finitely many simple submodules need not force a finite subcover.

Editorial extensions

If this is right

  • A compact divisor topology forces the socle to be finitely generated and essential, so the module is finitely cogenerated (Theorem 14).
  • If the compactness criterion holds, finite modules give compact divisor topologies, while every vector space of dimension at least two over an infinite field gives a noncompact one.
  • Pseudo-simple modules are exactly those whose divisor topology is $T_1$, Hausdorff, and discrete; for such modules all separation axioms reduce to one algebraic condition.
  • Uniserial modules are exactly those with linearly ordered open sets, so chains of cyclic submodules can be studied as nested topologies.
  • For factorial modules, the classes of irreducible elements form a dense set contained in every open dense set, which makes the divisor topology a Baire space.

Reading between the lines

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

  • The compactness theorem as stated needs an extra well-foundedness hypothesis: without it, a module such as $\mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})$ has exactly one simple submodule yet its divisor topology is noncompact, since the classes $[(p,0)]$ for infinitely many primes cannot be covered finitely.
  • The construction suggests a transfer principle: any module property decided by inclusion of cyclic submodules should be readable from the divisor topology, and the same topology could be defined for submodule lattices in other concrete categories.
  • A testable extension is to classify modules over a principal ideal domain for which the family $\{Rm : m \in M^{\#}\}$ is well-founded; in that setting the finite-socle compactness criterion would become a precise theorem.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces a divisor topology D(M) on the set EC(M^#) of equivalence classes of nonzero nongenerators of a module M over an integral domain, extending the divisor topology on domains to modules. It studies separation axioms, countability, Noetherianness, compactness, connectedness, and related properties of D(M), and it characterizes certain module classes (uniserial, simple, vector spaces, finitely cogenerated) in terms of D(M). It also introduces and studies pseudo-simple modules and uses them to characterize when D(M) is T1 or discrete.

Significance. The paper contains some correct and potentially useful local results, especially the pseudo-simple module concept, the T1 characterization in Theorem 12, and the Hausdorff characterization via the (*)-condition in Theorem 15. These parts could be of interest to researchers working on module-theoretic topologies. However, the paper's two headline structural theorems are false: Theorem 13, which is advertised as characterizing compactness of D(M) by finiteness of the family of minimal cyclic submodules or by finiteness of simple submodules, is contradicted by M=Z; and Theorem 4, characterizing Noetherianness of D(M), is contradicted by infinite-dimensional vector spaces and even by R^3 over an infinite field. Because the central claims are unsound, the paper in its current form does not establish its main advertised results.

major comments (3)
  1. [Section 4, Theorem 13] The converse direction of Theorem 13 is false. Take R=Z and M=Z. Then M^#=Z\{0,±1} and the family {Rm : m∈M^#}={kZ : |k|≥2} has no minimal elements, so the hypothesis "the family has only finitely many minimal elements" holds vacuously, and M has no simple submodules. Yet D(Z) is not compact: for any finite subfamily U_{n_1},...,U_{n_t} of the basis, choose a prime p that divides none of the n_i; then [p]∈EC(Z^#) is not contained in any U_{n_i}. Hence the claimed equivalence between compactness and finiteness of the family of minimal elements, and the claimed equivalence with finiteness of simple submodules, both fail. The proof's step "EC(M^#)=∪_{i=1}^n U_{m_i}" assumes that every cyclic submodule Rz contains one of the Rm_i; this hidden well-foundedness assumption is not stated and fails, for example, in M=Z⊕(Z/2Z), where the unique simple submodule is 0⊕(Z/2Z) but the cyclic submodule generated by (1,0) contains no minimal cyclic submodule.
  2. [Section 2, Theorem 4] The "if" direction of Theorem 4 is false. Let R be an infinite field and M=R^3. Then M is a vector space over the field R and is not simple. For nonzero m,n∈M, n|m iff m∈Rn, which happens exactly when n and m are scalar multiples, so U_m={[m]}. Thus D(M) is the discrete topology on the infinite set of one-dimensional subspaces of R^3, which is not Noetherian. The proof's appeal to [25, Theorem 6(iv)] is ineffective: in this discrete space every singleton is a maximal element of the basis, yet the space fails the descending chain condition on closed subsets because X, X\{x1}, X\{x1,x2}, ... is an infinite properly descending chain of closed sets.
  3. [Section 2, Theorem 2] The proof of Theorem 2 contains two false equalities. Density of EC(Irr(M)) requires showing EC(M^#) ⊆ closure(EC(Irr(M))), not EC(M^#) ⊆ EC(Irr(M)). For a factorial module, from m=r_1...r_n m' one obtains [m]∈closure{[m']}, which does not imply [m]∈EC(Irr(M)); indeed for M=Z, EC(Irr(M)) consists of classes of primes while EC(Z^#) contains classes such as [6]. The line "take an open dense set O in D(M) then O=EC(M^#)" is also false in general. The theorem's conclusion may still be true, but the proof as written is invalid and needs to be replaced by an argument showing that every dense open set meets each singleton {[m]} for m∈Irr(M).
minor comments (5)
  1. [Remark 1 and Theorem 13] Remark 1 says the paper assumes M is not a simple module unless otherwise specified, but Theorem 13 and Theorem 4 do not state this assumption. As written, Theorem 13's equivalence between finitely many minimal elements of {Rm : m∈M^#} and finitely many simple submodules fails for simple M, since the family is empty but M has one simple submodule. Please add the non-simple assumption explicitly in all statements where it is needed.
  2. [Example 4(iv)] Example 4(iv) contains a garbled sentence: "Z6 is a pseudo simple module since Z2 = Z4 and Z3 are simple submodules." Z4 is not a submodule of Z6; the intended statement is that the cyclic submodules generated by 2 and 4 are both Z2, and the submodule generated by 3 is Z3.
  3. [Example 10] The notation D(Z^#_{p^α}) in Example 10 is confusing; it should likely be D(Z_{p^α}).
  4. [Theorem 9 proof] In the proof of Theorem 9, from maximality of ann((m,m'))=ann(m)∩ann(m') and the inclusion ann(m)∩ann(m')⊆ann(m), the conclusion ann(m)=ann(m)∩ann(m') requires the additional observation that ann(m)≠R; the argument as written skips this justification.
  5. [Throughout] There are several minor typos: "extesion" for "extension", "seperated" for "separated", and the equivalence relation definition says "for every m,n∈M" where it should say "for every m,n∈M^#".

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the topological characterizations are derived from the definitions, with only a minor proof deferral to prior work by overlapping authors.

full rationale

The paper's core construction, defining a topology on EC(M#) via the basis {U_m} from divisibility, is developed for modules from the definitions and then used to prove the separation, countability, compactness, and Noetherian results. The only load-bearing deferred item is Proposition 1, where the basis verification is omitted as 'analogous to [25, Proposition 1]'; this is a proof deferral to a prior paper by overlapping authors, not an assumption of the theorem being proved. Theorem 12 and Theorem 17 genuinely prove the equivalence between pseudo-simplicity and T1/discreteness rather than defining pseudo-simple modules as 'the property that makes D(M) T1.' Theorem 13, the compactness characterization, contains a substantive mathematical gap in its converse, but that is a correctness defect rather than a circular reduction: the conclusion is not already contained in the hypothesis by construction. There are no fitted parameters, no renamed empirical patterns, and no self-citation chain that forces the headline result. Score 1 reflects only the minor self-citation and omitted analogous proof, which do not infect the derivation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper uses standard ZFC, commutative algebra and general topology. No free parameters are fitted. The new definition 'pseudo simple module' is a mathematical concept, not an invented entity in the sense of a new physical or formal object. The main hidden assumption is the implicit well-foundedness of cyclic submodules in Theorem 13, noted as a red flag.

assumptions (3)
  • domain assumption R is an integral domain and M is a nonzero unital module
    Stated at the start of Section 1; all results assume this.
  • standard math The family {U_m} is a basis for a topology on EC(M#)
    Proposition 1, proof omitted and deferred to [25]; the basis property is standard but not demonstrated in this paper.
  • standard math Standard results from commutative algebra and general topology, e.g., Munkres [16]
    Used throughout, including the Baire space lemma and Noetherian space criteria.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Divisor Topology of Modules over Domains." pith.science (2026). https://pith.science/paper/7UNAYW4S

@misc{pith2026250601179,
  author       = {Pith},
  title        = {Pith review of: On Divisor Topology of Modules over Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UNAYW4S}},
  note         = {Machine review of arXiv:2506.01179}
}
abstract

Let $M\ $be a module over a domain $R$ and $M^{\#}=\{0\neq m\in M:Rm\neq M\}$ be the set of all nonzero nongenerators of $M.\ $Consider following equivalence relation $\sim$ on $M^{\#}$ as follows: for every $m,n\in M^{\#},\ m\sim n$ if and only if $Rm=Rn.\ $Let $EC(M^{\#})$ be the set of all equivalence classes of $M^{\#}$ with respect to $\sim$. In this paper, we construct a topology on $EC(M^{\#})$ which is called divisor topology of $M\ $and denoted by $D(M).$ Actually, $D(M)$ is extension of the divisor topology $D(R)$ over domains in the sense of Yi\u{g}it and Koc to modules. We investigate separation axioms $T_{i}$ for every $0\leq i\leq5,$ first and second countability, connectivity, compactness, nested property, and Noetherian property on $D(M)$. Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of $D(M)$. Furthermore, we prove that $D(M)$ is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when $D(M)$ is a discrete space.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [6]

    Atiyah, M.F., Macdonald, I.G. (1969). Introduction to Commutative Algebra. Addison- Wesley Publishing Co., Reading, Mass., London-Don Mills, Ontario

    work page 1969

  2. [1]

    Ahmed, H. (2018). S-Noetherian spectrum condition. Communications in Algebra, 46(8), 3314-3321

    work page 2018

  3. [2]

    D., Arabaci, T., Tekir, ¨U., & Ko¸ c, S

    Anderson, D. D., Arabaci, T., Tekir, ¨U., & Ko¸ c, S. (2020). On S-multiplication modules. Communications in Algebra, 48(8), 3398-3407

    work page 2020

  4. [3]

    W., & Fuller, K

    Anderson, F. W., & Fuller, K. R. (1992). Rings and categories of modules (Vol. 13). Springer Science & Business Media

    work page 1992

  5. [4]

    D., & Winders, M

    Anderson, D. D., & Winders, M. (2009). Idealization of a module. Journal of commutative algebra, 1(1), 3-56

    work page 2009

  6. [5]

    Arenas, F. G. (1999). Alexandroff spaces. Acta Math. Univ. Comenianae, 68(1), 17-25

    work page 1999

  7. [7]

    Barnard, A. (1981). Multiplication modules. J. Algebra., 71(1), 174-178

    work page 1981

  8. [8]

    Callialp, F., Ulucak, G., & Tekir, ¨U. (2017). On the Zariski topology over an L-module M . Turkish Journal of Mathematics, 41(2), 326-336

    work page 2017

Show all 25 references
  1. [9]

    (2023, December)

    C ¸ eken, S. (2023, December). Dual Zariski Spaces of Modules. In Algebra Colloquium (Vol. 30, No. 04, pp. 569-584). World Scientific Publishing Company

    2023

  2. [10]

    C ¸ eken, S. (2022). On S-second spectrum of a module. Revista de la Real Academia de Ciencias Exactas, F ´ ısicas y Naturales. Serie A. Matem´ aticas, 116(4), 171

    2022

  3. [11]

    A., & Smith, P

    El-Bast, Z. A., & Smith, P. F. (1988). Multiplication modules. Communications in Algebra, 16(4), 755-779

    1988

  4. [12]

    Facchini, A., & Salce, L. (1990). Uniserial modules: sums and isomorphisms of subquotients. Communications in Algebra, 18(2), 499-517

    1990

  5. [13]

    Hartshorne, R. (2013). Algebraic Geometry, vol. 52. Springer

    2013

  6. [14]

    Lu, C. P. (1977). Factorial modules. The Rocky Mountain Journal of Mathematics, 7(1), 125-139

    1977

  7. [15]

    Kim, C. (2019). The Zariski topology on the prime spectrum of a commutative ring (Doctoral dissertation, California State University, Northridge)

    2019

  8. [16]

    Munkres, J. (2000). Topology, 2nd edn. Prentice Hall, Upper Saddle River

    2000

  9. [17]

    Moghaderi, J., & Nekooei, R. (2010). Valuation, discrete valuation and Dedekind modules. International Electronic Journal of Algebra, 8(8), 18-29

    2010

  10. [18]

    and Pendleton, R

    Ohm, J. and Pendleton, R. (1968). Rings with Noetherian spectrum, Duke Math. J. 35, 631–639. ON DIVISOR TOPOLOGY OF MODULES OVER INTEGRAL DOMAINS 17

    1968

  11. [19]

    A., Tekir, ¨U., & Ko¸ c, S

    ¨Ozen, M., Naji, O. A., Tekir, ¨U., & Ko¸ c, S. (2023). On Modules Satisfying S-Noetherian Spectrum Condition. Communications in Mathematics and Statistics, 11(3), 649-662

    2023

  12. [20]

    Richmond, T. (2020). General Topology: An Introduction. De Gruyter

    2020

  13. [21]

    Sharp, R. Y. (2000). Steps in commutative algebra (No. 51). Cambridge university press

    2000

  14. [22]

    A., Seebach, J

    Steen, L. A., Seebach, J. A., & Steen, L. A. (1978). Counterexamples in topology (Vol. 18). New York: Springer

    1978

  15. [23]

    Tuganbaev, A. A. (2009). Bezout modules and rings. Journal of Mathematical Sciences, 163(5), 596-597

    2009

  16. [24]

    A., Tekir, ¨U., & Ko¸ c, S

    Yıldız, E., Ersoy, B. A., Tekir, ¨U., & Ko¸ c, S. (2021). On S-Zariski topology. Communications in Algebra, 49(3), 1212-1224

    2021

  17. [25]

    Yi˘ git, U., & Ko¸ c, S. (2025). On divisor topology of commutative rings. Ricerche di Matem- atica, 1-13. Department of Mathematics, Marmara University, Istanbul, Turkey. Email address : utekir@marmara.edu.tr Department of Mathematics, Istanbul Medeniyet University, 34700,¨Us...

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.