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Some generalizations of strongly prime ideals

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Pith's one-line read Strongly prime ideals extend to two 2-absorbing classes

desk verdict New definitions and easy consequences are fine, but the main intersection theorem and the characterization in Theorem 3.3 are not proved as written. read the letter →

arxiv 1908.06744 v1 pith:MD7JBGKR submitted 2019-08-19 math.AC

H. Ansari-Toroghy , F. Farshadifar , S. Maleki-Roudposhti This is my paper · ORCID
classification math.AC MSC 13G0513C1313A15
keywords stronglyprimeideal2-absorbingprimarysubmodulerootydomainmultiplicationmodule
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 introduces two new classes of ideals—strongly 2-absorbing primary ideals and strongly 2-absorbing ideals—as generalizations of strongly prime ideals, moving the defining "either factor belongs to the ideal" condition from two factors to three factors in the quotient field. The motivating claim is that the new classes sit naturally between strongly prime ideals and their ordinary 2-absorbing analogues: every strongly primary ideal is strongly 2-absorbing primary, every strongly prime ideal is strongly 2-absorbing, and in rooty domains (domains whose prime ideals are strongly radical) the new conditions admit clean characterizations in terms of inverse multiplication by quotient-field elements. The paper also proves closure properties, including intersections of strongly prime ideals and, in one of its main results, intersections of strongly primary ideals, and derives finiteness of minimal strongly 2-absorbing submodules for Noetherian modules. A sympathetic reader would care because these constructions supply a bridge between classical prime/primary decomposition and the newer 2-absorbing framework.

What carries the argument

The load-bearing object is the quotient field $K$ of the integral domain $R$: the new definitions quantify over $x,y,z\in K$ rather than over elements of $R$, so the ideal membership test applies even to field elements that lie outside $R$. The main working identity is the inverse-containment criterion of Theorem 3.3: a 2-absorbing ideal $I$ is strongly 2-absorbing exactly when, for every $x,y\in K$ with $xy\notin R$, one has $x^{-1}I\subseteq I$ or $y^{-1}I\subseteq I$. For the primary version, the companion mechanism is the set $E(I)$ of field elements none of whose positive powers belong to $I$, which converts "some power enters $I$" conclusions into a separation condition.

What would settle it

Test the key step of Theorem 2.9 in a concrete non-rooty domain such as $R=K[[X^2,X^3]]$, the ring used in Example 2.3: choose nonzero strongly primary ideals $P,Q$ and elements $x,y,z$ in the quotient field with $xyz\in P\cap Q$, and check whether $xy\in P\cap Q$ or some power of $yz$ or $xz$ lies in $P\cap Q$. If all three alternatives fail while $xyz\in P\cap Q$, the theorem as stated is false; if none fail, the intersection claim may survive even though its proof needs repair.

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Extended reading notes

Core claim

The central discovery is that the strong-prime mechanism—checking membership of the factors of a product taken from the quotient field—can be transplanted to the 2-absorbing setting in two versions. For a 2-absorbing ideal $I$, the paper characterizes strong 2-absorbingness by a ratio-symmetric condition: for every $x,y$ in the quotient field $K$ with $xy\notin R$, either $x^{-1}I\subseteq I$ or $y^{-1}I\subseteq I$ (Theorem 3.3). For strongly 2-absorbing primary ideals, it gives a parallel characterization under a rooty-domain hypothesis using the set $E(I)=\{x\in K:x^n\notin I \text{ for all } n\ge 1\}$ (Theorem 2.5). Along the way it proves that strongly prime ideals are strongly 2-absorbing, strongly primary ideals are strongly 2-absorbing primary, valuation domains make every 2-absorbing ideal strongly 2-absorbing, and Noetherian modules have finitely many minimal strongly 2-absorbing submodules.

Load-bearing premise

The load-bearing premise in the proof of Theorem 2.9 is that an ideal of $R$ is closed under multiplication by arbitrary elements of the quotient field: from $z^n\in P$ and $y^t\in Q$ the proof infers $(xy)^{nt}\in P\cap Q$, which requires exactly that closure, and that closure fails for a general integral domain.

Editorial extensions

If this is right

  • Every strongly prime ideal is strongly 2-absorbing, and every strongly primary ideal is strongly 2-absorbing primary, so domains that already have strongly prime ideals automatically acquire both new classes (Propositions 3.2 and 2.2).
  • In valuation domains, every 2-absorbing ideal is strongly 2-absorbing, so the new condition imposes no extra restriction there (Proposition 3.11).
  • Theorem 3.3 turns the definition into a two-variable test: to check whether a 2-absorbing ideal is strongly 2-absorbing, it suffices to test inverse containment for pairs $x,y\in K$ with $xy\notin R$.
  • The intersection of two nonzero strongly prime ideals is strongly 2-absorbing (Theorem 3.16), producing non-prime examples of strongly 2-absorbing ideals.
  • Every Noetherian module contains only finitely many minimal strongly 2-absorbing submodules (Theorem 3.30), so the class shares the finiteness behavior familiar from primary decomposition.

Reading between the lines

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

  • Editorial inference: if Theorem 2.9 can be repaired under an added hypothesis such as rootiness or strong radicality, strongly 2-absorbing primary ideals would be closed under finite intersections, giving a 2-absorbing analogue of primary decomposition.
  • Editorial inference: the ratio-symmetric criterion of Theorem 3.3 suggests that strongly 2-absorbing ideals should behave well under passage to overrings, and Proposition 3.20 already shows extension to overrings; a natural next question is whether this extension is unique or canonical.
  • Editorial inference: because the primary version's characterization requires $K\setminus E(I)$ to be closed under addition (true in rooty domains), one could test whether the class of strongly 2-absorbing primary ideals in non-rooty domains coincides with some valuation-theoretic condition on value groups.
  • Editorial inference: the finite-minimal-submodules theorem suggests a decomposition question: can every strongly 2-absorbing submodule of a Noetherian module be expressed as an intersection of minimal ones, mirroring primary decomposition?
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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

4 major / 5 minor

Summary. The paper introduces two new classes of ideals: strongly 2-absorbing primary ideals and strongly 2-absorbing ideals, together with their submodule analogues. These are defined by extending the strong-prime condition to products of three elements in the quotient field of an integral domain. The paper claims that these classes generalize strongly prime ideals, and proves several basic properties: every strongly primary ideal is strongly 2-absorbing primary (Proposition 2.2), every strongly prime ideal is strongly 2-absorbing (Proposition 3.2), characterizations are given in Theorems 2.5 and 3.3, intersections of two strongly primary ideals are claimed to be strongly 2-absorbing primary (Theorem 2.9), and similarly for strongly prime ideals (Theorem 3.16). Localization, chain conditions, and minimal submodule results are also stated.

Significance. If the results are correct, the paper offers a reasonable common framework for strongly prime and 2-absorbing notions, and the submodule analogues are potentially useful. The definitional work is natural, and the simple implications (Propositions 2.2 and 3.2) are correct, as are several of the concrete examples. However, the main structural theorems, especially the intersection theorems and the key characterizations, rest on proofs that contain invalid steps. The paper also leaves Theorem 3.16 unproved except for a reference to the flawed proof of Theorem 2.9. The overall value of the paper will depend on whether the stated results can be supplied with correct proofs.

major comments (4)
  1. [§2, Theorem 2.9] The proof is invalid. In the final step, the text claims that if z^n ∈ P and y^t ∈ Q, then (xy)^{nt} ∈ P and (xy)^{nt} ∈ Q 'by definition of an ideal.' This assumes that P and Q are closed under multiplication by arbitrary elements of the quotient field K, which is false since P and Q are ideals of R. Concretely, in R = F[[X^2,X^3]], take P = X^2F[[X]], Q = X^3F[[X]], x = X^{-1}, y = X^2, z = X^2. Then z ∈ P, y^2 ∈ Q, and xyz = X^3 ∈ P ∩ Q, but (xy)^2 = X^2 ∉ Q. Thus the inference is false. Since Theorem 2.9 is used in Proposition 2.12 and Proposition 3.25(c), and Theorem 3.16 is asserted to be 'similar', the intersection theorems are not established as written and require a completely new proof.
  2. [§3, Theorem 3.3] The proof of (b) ⇒ (a) contains an unjustified step. The text says that if xy ∈ R, xz ∈ R, and yz ∈ R, then 'we are done since I is a 2-absorbing ideal of R.' But the 2-absorbing condition applies only to triples a,b,c ∈ R, and the given x,y,z are elements of K that need not lie in R. This is not a harmless oversight: for R = Z and I = 6Z, I is 2-absorbing, and with x = 2/3, y = 3/2, z = 6 we have xyz = 6 ∈ I, while xy = 1, xz = 4, yz = 9 all lie in R and none lies in I. The proof must use condition (b) to handle this case, but no such argument is provided. This leaves the main characterization of strongly 2-absorbing ideals unproved, and all later results that invoke Theorem 3.3 are therefore conditional.
  3. [§2, Theorem 2.7(a)] The proof's case analysis is not correct. After applying the strong 2-absorbing primary condition to (xy/ab)(a/x)(b/1) = y, one of the resulting alternatives is (ab/x)^n ∈ I. The proof then says 'x(ab/x) ∈ xR ⊆ I', but x(ab/x) = ab, and ab was chosen outside I. Multiplication by x ∈ I does not put ab in I unless ab/x ∈ R, which is not known. Since the argument for I^2 ⊆ J ∪ H relies on ruling out or using this case, the proof of Theorem 2.7(a) is incomplete. This also affects Corollary 2.8 and the parallel proof of Theorem 3.18.
  4. [§3, Proposition 3.15(a)] The proof does not justify the key implication. It states that for x ∈ K \ R, the strong semiprime property gives x^2 ∉ P, and then 'Since P is strongly 2-absorbing, this implies that x^{-1}P ⊆ P by Theorem 3.3.' However, Theorem 3.3(b) applies to a pair x,y with xy ∉ R; taking y = x only gives x^2 ∉ P, not x^2 ∉ R, and the 'either' clause in (b) does not force x^{-1}P ⊆ P. The argument would need a different choice of the second element or a separate proof, which is not supplied. Thus the conclusion that strongly semiprime plus strongly 2-absorbing implies strongly prime is not established.
minor comments (5)
  1. [§3, before Example 3.5] The sentence 'Example 3.5, Proposition 3.6, and Example 3.7 show that the converse of Proposition 2.2 is not true in general' should refer to Proposition 3.2, not Proposition 2.2, since the examples concern strongly 2-absorbing ideals and strongly prime ideals.
  2. [§3, Example 3.7] The text says 'a strongly 2-absorbing prime ideal'; the paper's terminology defines 'strongly 2-absorbing ideal' and 'strongly 2-absorbing primary ideal', but not 'strongly 2-absorbing prime ideal'. This is likely a typo and should be corrected to avoid confusion.
  3. [§3, Proposition 3.11] In the proof, the line 'If x, y, and z are in V, we are done' should explicitly say that the conclusion follows from the 2-absorbing property of I, since the strong 2-absorbing condition reduces to the ordinary 2-absorbing condition when all three factors lie in V.
  4. [§1, references] Several references are incomplete or have typographical issues: [1] gives only a page number '111' without the article's full page range, and [9] writes '441451' instead of '441–451'. These should be checked against the published versions.
  5. [§2, Notation 2.4] The notation E(S) is used before the definition of 'strongly radical' is recalled, and the connection is not immediately clear. A short explanation of the role of E(I) in the definition of strongly 2-absorbing primary would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper defines new classes from external prior notions and proves properties directly.

full rationale

The paper introduces strongly 2-absorbing primary ideals and strongly 2-absorbing ideals as new definitions built on the external, cited concepts of strongly prime ideals, strongly primary ideals, and 2-absorbing ideals. The main results, such as Theorem 2.5 and Theorem 3.3, are proved directly from these definitions and from cited background results, without fitting parameters or assuming the target conclusion. There is no self-citation chain carrying the argument: the cited references are prior works by other authors, not load-bearing appeals to the present authors' own unverified results. The intersection theorems (Theorem 2.9 and Theorem 3.16) contain an invalid ideal-closure step in the proof, as the skeptic notes, but that is a proof error, not a circularity: the statement is not assumed, and the argument does not reduce to its inputs by construction. No prediction is fitted, no known result is merely renamed, and no uniqueness theorem is imported from the authors' own prior work. The paper is self-contained against external benchmarks and its definitions are genuinely new generalizations, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 4 invented entities

The paper is a standard commutative algebra paper: it introduces new definitions (the four invented entities above) and proves their properties using the standing assumption that R is an integral domain, standard set theory (Zorn's lemma), and cited results on ideals, modules, and valuations. There are no numerical free parameters.

assumptions (5)
  • domain assumption R is an integral domain with quotient field K throughout.
    This is the standing framework stated in the introduction; all definitions and proofs operate over an integral domain and its quotient field.
  • standard math Zorn's lemma is used to prove existence of maximal elements in Lemma 3.29 and Corollary 3.22.
    Used without proof to show every strongly 2-absorbing submodule contains a minimal one and to form a largest strongly 2-absorbing ideal in chained rings.
  • domain assumption In Theorem 2.5, the additional assumption that K \ E(I) is closed under addition (or that R is rooty) is required for (b) => (a).
    This is a stated hypothesis, not a general fact; it restricts the domains for which the characterization holds.
  • standard math Valuation structure of DVRs and formal power series rings used in examples (e.g., K[[X^2,X^3]]) is taken as known.
    The examples compute valuations of elements in K((X)) and use properties of power series rings without proof.
  • standard math In Proposition 3.24, the identity for colon ideals in multiplication modules is cited from El-Bast and Smith.
    The paper relies on a known result about sums of colon ideals for faithful finitely generated multiplication modules over chained rings.
invented entities (4)
  • Strongly 2-absorbing primary ideal
    purpose: Generalizes strongly prime and strongly primary ideals to products of three elements, allowing powers of two pairwise products.
    This is a new definition introduced in Definition 2.1. It has no falsifiable handle outside the paper; its utility rests on the properties proved here.
  • Strongly 2-absorbing ideal
    purpose: Generalizes strongly prime ideals to a 2-absorbing ideal that behaves well with three elements from the quotient field.
    New definition in Definition 3.1, similar to the above.
  • Strongly 2-absorbing primary submodule
    purpose: Module-theoretic analogue of strongly 2-absorbing primary ideals, defined via colon ideals.
    Definition 2.11; it is a bookkeeping version of the ideal class for modules.
  • Strongly 2-absorbing submodule
    purpose: Module-theoretic analogue of strongly 2-absorbing ideals, defined via colon ideals.
    Definition 3.23; it is a bookkeeping version of the ideal class for modules.

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Cite this review

Pith. "Pith review of Some generalizations of strongly prime ideals." pith.science (2026). https://pith.science/paper/MD7JBGKR

@misc{pith2026190806744,
  author       = {Pith},
  title        = {Pith review of: Some generalizations of strongly prime ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MD7JBGKR}},
  note         = {Machine review of arXiv:1908.06744}
}
read the original abstract

In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these classes of ideals.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    Anderson and D.F

    D.D. Anderson and D.F. Anderson, Multiplicatively closed subsets of fields , Houston J. Math. 13 (1989), 111

    work page 1989

  2. [2]

    Anderson and J

    D.F. Anderson and J. Park, Rooty and root-closed domains , Advances in Commutative ring theory, Lecture Notes Pure Applied Math., Marcel Dekker 205 (1999), 8799

    work page 1999

  3. [3]

    Badawi, On 2-absorbing ideals of commutative rings , Bull

    A. Badawi, On 2-absorbing ideals of commutative rings , Bull. Austral. Math. Soc. 75 (2007), 417-429

    work page 2007

  4. [4]

    Badawi and E.G

    A. Badawi and E.G. Houston, Powerful ideals, strongly primary ideals, almost pseudo- valuation domains, and conductive domains , Comm. Algebra, 30 (2002), 1591-1606

    work page 2002

  5. [5]

    Badawi, U

    A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings , Bull. Korean Math. Soc., 51 (4) (2014), 1163-1173

    work page 2014

  6. [6]

    Barnard, Multiplication modules, J

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

    work page 1981

  7. [7]

    El-Bast and P.F

    Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra 16 (1988), 755–779

    work page 1988

  8. [8]

    Hedstrom and G

    J. Hedstrom and G. Houston, Pseudo-valuation domains , Pacific J. Math. 75 (1) (1978), 137–147

    work page 1978

Show all 10 references
  1. [9]

    Sato and T

    J. Sato and T. Sugatani, On the radical ideal of seminormal rings , Comm. Algebra 18 (1990), 441451

    1990

  2. [10]

    Sharp, Step in commutative algebra , Cambridge University Press, 1990

    R.Y. Sharp, Step in commutative algebra , Cambridge University Press, 1990. * Department of pure Mathematics, F aculty of mathematical S ciences, University of Guilan, P. O. Box 41335-19141, Rasht, Iran E-mail address : ansari@guilan.ac.ir ** (Corresponding Author) Assistant P...

    1990

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