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REVIEW 4 major objections 9 minor 33 references

$(u,v)$-absorbing primary hyperideals in multiplicative hyperrings

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Pith's one-line read The paper defines $(u,v)$-absorbing primary hyperideals in commutative multiplicative hyperrings, proves that their radical is prime for $C$-hyperideals, and characterizes the class in Dedekind and divided hyperdomains.

desk verdict A legitimate new definition in a niche subfield, but the central structural theorem has a real proof gap and one example is arithmetically wrong; worth refereeing, not worth trusting as is. read the letter →

arxiv 2506.02989 v1 pith:QWOYOIKF submitted 2025-06-03 math.AC

Mahdi Anbarloei This is my paper
classification math.AC MSC 20N2016Y20
keywords multiplicativehyperring(uv)-absorbingprimaryhyperidealprimeradicalC-hyperidealhyperdomainlocal
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

Commutative multiplicative hyperrings replace ordinary multiplication by a set-valued hyperoperation, and this paper transplants the ring notion of $(u,v)$-absorbing primary ideals into them. A proper hyperideal $P$ is $(u,v)$-absorbing primary when a product of $u$ non-units contained in $P$ forces either the first $v$ factors into $P$ or the remaining factors into the radical $\operatorname{rad}(P)$. The paper aims to show this class sits strictly between $(u,v)$-absorbing prime hyperideals and primary hyperideals: every prime version is a primary version, the converse fails by example, and for $C$-hyperideals the radical of the new object is always prime. It then proves two characterization results: in a Dedekind hyperdomain a $C$-hyperideal is $(u,v)$-absorbing primary exactly when its radical is prime, and in a divided hyperring the new class is just the primary hyperideals. A further structural theorem says that a strong $C$-hyperideal that is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary without being $(u,v)$-absorbing primary can exist only when the whole hyperring is local.

What carries the argument

The load-bearing mechanism is the $C$-hyperideal condition together with the prime radical. A $C$-hyperideal is one for which a finite hyperproduct that merely meets the hyperideal is forced to lie entirely inside it, and a strong $C$-hyperideal does the same for finite sums of hyperproducts; this is what lets the proofs pass from a single containment to radical membership and back. The radical tracks the primary half of the definition, and Theorem 2.7 uses the $C$-condition to show that $\operatorname{rad}(P)$ inherits primality. In the Dedekind hyperdomain result, invertibility of hyperideals supplies maximality of the prime radical; in the locality theorem, the strong $C$-property on $\operatorname{rad}(P)$ is used to deduce that the sum of a non-unit and a unit is a unit, which is the criterion for a local hyperring.

What would settle it

Compute $\operatorname{rad}(P\circ M)$ in a concrete local multiplicative hyperring such as the $\mathbb{Z}_\Phi$ construction with $\Phi=\{2,3\}$, taking $P$ a prime hyperideal and $M$ the unique maximal hyperideal; if the radical is strictly larger than $P$, or if $P\circ M$ fails the $(u,v)$-absorbing primary condition, Proposition 2.8 is false. For Theorem 2.18, look for a Dedekind hyperdomain with a $C$-hyperideal whose radical is prime but which is not primary, which would refute the claimed equivalence.

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Extended reading notes

Core claim

The central claim is that Definition 2.1 gives a useful intermediate class in commutative multiplicative hyperrings. For integers $u>v$, a proper hyperideal $P$ is $(u,v)$-absorbing primary if $x_1\circ\cdots\circ x_u\subseteq P$ with $x_1,\ldots,x_u\notin U(A)$ implies $x_1\circ\cdots\circ x_v\subseteq P$ or $x_{v+1}\circ\cdots\circ x_u\subseteq \operatorname{rad}(P)$. The paper establishes that every $(u,v)$-absorbing prime hyperideal is $(u,v)$-absorbing primary and that the reverse fails (Example 2.4), and that the primary property is monotone in both parameters. Its main structural assertions are Theorem 2.7, that the radical of a $(u,v)$-absorbing primary $C$-hyperideal is prime; Theorem 2.12, that a strong $C$-hyperideal which is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary but not $(u,v)$-absorbing primary forces the hyperring to be local; and Theorem 2.18, that in a Dedekind hyperdomain a $C$-hyperideal is $(u,v)$-absorbing primary if and only if its radical is prime. Proposition 2.8 further claims that in a local hyperring whose hyperideals are $C$-hyperideals, the product $P\circ M$ of a prime hyperideal with the maximal hyperideal is $(u,v)$-absorbing primary.

Load-bearing premise

The load-bearing assumption is that inside a local multiplicative hyperring, the radical of the product $P\circ M$ of a prime hyperideal $P$ with the maximal hyperideal $M$ is exactly $P$; the proof of Proposition 2.8 uses this equality without proving it, and the proposition collapses if it is false.

Editorial extensions

If this is right

  • Every $(u,v)$-absorbing prime hyperideal is automatically $(u,v)$-absorbing primary, and Example 2.4 shows the converse fails, so the primary version is strictly broader.
  • For a $(u,v)$-absorbing primary $C$-hyperideal, the radical is a prime hyperideal (Theorem 2.7), matching the radical behaviour of classical primary ideals.
  • In a Dedekind hyperdomain, a $C$-hyperideal is $(u,v)$-absorbing primary if and only if its radical is prime (Theorem 2.18).
  • In a divided multiplicative hyperring, the $(u,v)$-absorbing primary $C$-hyperideals are exactly the primary hyperideals (Theorem 2.19).
  • A strong $C$-hyperideal that is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary but not $(u,v)$-absorbing primary can exist only in a local multiplicative hyperring (Theorem 2.12).

Reading between the lines

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

  • As an extension beyond the paper, Definition 4.1's $(u,v)$-absorbing $I$-primary variant could be tested against the Section 2 theorems; replacing $x_1\circ\cdots\circ x_u\subseteq P$ by $x_1\circ\cdots\circ x_u\subseteq P\setminus IP$ would show how much of the hierarchy survives.
  • A natural converse question the paper does not ask is whether every local multiplicative hyperring contains a $(u+1,v+1)$- or $(u+1,v)$-absorbing primary strong $C$-hyperideal that is not $(u,v)$-absorbing primary; Theorem 2.12 only proves the one-way implication.
  • The scope of Proposition 2.8 hinges on the unproved equality $\operatorname{rad}(P\circ M)=P$; proving or disproving that equality for arbitrary local multiplicative hyperrings would decide whether the product construction is as broad as stated.
  • If the new class is genuinely distinct only outside divided hyperrings, a concrete test is to check whether the $\mathbb{Z}_\Phi$ construction with $\Phi=\{2,3\}$ has $(u,v)$-absorbing primary hyperideals whose radicals are prime but which are not primary.
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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 / 9 minor

Summary. The paper introduces (u,v)-absorbing primary hyperideals in commutative multiplicative hyperrings, a notion intended to interpolate between primary hyperideals and the author's earlier (u,v)-absorbing prime hyperideals. The main results are: the radical of a (u,v)-absorbing primary C-hyperideal is prime (Theorem 2.7); a sufficient condition for P∘M to be such a hyperideal in a local hyperring (Proposition 2.8); a locality criterion when a stronger absorbing property is not inherited (Theorem 2.12); intersection and colon properties (Propositions 2.9 and 2.15); characterizations in Dedekind hyperdomains (Theorem 2.18) and divided multiplicative hyperrings (Theorem 2.19); and stability under homomorphisms, quotients, localizations, and matrix hyperrings (Theorems 2.20-2.24). The paper also presents examples separating the new class from (u,v)-absorbing prime hyperideals.

Significance. If the main structural claims are correct, the definition is a natural extension of the ring-theoretic (u,v)-absorbing primary ideals to multiplicative hyperrings, and the paper provides a useful collection of tools (intersections, colons, residues, localization, and matrix constructions) for working with this class. The paper's strengths are its clean definition, its explicit use of the C-hyperideal and strong C-hyperideal machinery, and its systematic treatment of standard hyperring constructions. However, the current version contains several load-bearing gaps: the proof of Theorem 2.7 relies on an unjustified uniform-exponent step, Proposition 2.8 invokes an unproved radical equality, Example 2.16 contains an arithmetic error, and Theorem 2.18 omits the zero-ideal case and an unproved maximality assertion. These issues affect the paper's central hierarchy claim and must be repaired before the results can be regarded as sound. There is no computational verification; the examples are hand-checked and several have notational or arithmetic problems.

major comments (4)
  1. [Section 2, Theorem 2.7] The proof of Theorem 2.7 contains an unjustified step: from a∘b⊆rad(P) the author concludes 'a^n∘b^n⊆P for some n∈N'. For a C-hyperideal, rad(P) is the set of elements some power of which lies in P, but this provides one exponent per element of a∘b and no uniform bound on those exponents. Moreover, a^n∘b^n consists of products of n elements chosen from a and n from b, not powers of single elements of a∘b, so the inclusion does not follow from the definition. The subsequent application of Definition 2.1 with x1=...=xv−1=a, x_v=x, x_{v+1}=...=x_{u−1}=b, and x_u=y depends entirely on this step. Since Theorem 2.7 is used in the proofs of Theorem 2.14, Proposition 2.15, Theorem 2.18, and Theorem 2.19, this gap threatens the paper's central structural assertion. The proof needs either a correct argument producing a uniform exponent, or an additional hypothesis (for example, a Noetherian-like finiteness condition).
  2. [Section 2, Proposition 2.8] The proof of Proposition 2.8 uses the equality P = rad(P∘M) without proof, where P is a prime hyperideal and M is the unique maximal hyperideal. In classical commutative rings this equality can be shown when P⊆M, but the hyperring analogue is not automatic and the paper provides neither a proof nor a citation. If the equality fails, the step 'xv+1∘...∘xu⊆P=rad(P∘M)' collapses, and the claim that P∘M is (u,v)-absorbing primary is unsupported. Please either prove the radical equality from the hyperring axioms or state explicitly the hypotheses under which it holds.
  3. [Section 2, Example 2.16] Example 2.16 asserts ⟨150⟩=⟨3⟩∩⟨5⟩∩⟨7⟩ in the multiplicative hyperring Z_Φ with Φ={2,4}. In the underlying ring Z, the intersection of these ideals is ⟨105⟩ (150 is not divisible by 7), so the example is arithmetically incorrect. Since the example is meant to show that the equal-radical condition in Proposition 2.15 is crucial, it does not support that claim as written. In addition, the hyperring Z_Φ with Φ={2,4} does not have an identity element in the sense of the paper (an element e with a∈e∘a for every a), because no integer e satisfies 1∈{2e,4e}; this conflicts with the standing assumption that A has identity element 1. Similar concerns affect other examples using Φ not containing 1. The example should be recalculated or replaced.
  4. [Section 2, Theorem 2.18] The reverse implication in Theorem 2.18 uses the assertion 'Since rad(P) is prime, by the hypothesis, rad(P) is maximal.' The paper does not prove that every nonzero prime hyperideal in a Dedekind hyperdomain is maximal, and the statement is false for P=0, where rad(P)=0 is not maximal unless the hyperdomain is a hyperfield. The later step ⟨y, rad(P)⟩=A for y∉rad(P) depends on maximality. Please supply the missing argument for Dedekind hyperdomains or explicitly exclude the zero ideal and justify the maximality claim.
minor comments (9)
  1. [Title/Abstract] The title contains the typo 'MUL TIPLICA TIVE', and the introduction contains 'hyerrings' and 'hperrings'; these should be corrected.
  2. [Introduction] The introduction states u,v∈Z, while the rest of the paper uses u,v∈N with u>v; the definition should be stated consistently with N.
  3. [Section 2, first paragraph] The first sentence of Section 2 contains the typo 'A proper hyperideal ofPofAis said to be...' and should read 'A proper hyperideal P of A is said to be...'.
  4. [Example 2.2] In Example 2.2(i), the description of P=2Z[i] as a set of the form {-2x-2yi, 6x+6yi : x,y∈Z} is confusing and likely misprinted; the usual description is {2a+2bi : a,b∈Z}. In Example 2.2(ii), α and β are said to be in Z, but the hyperring is A=Z+3xZ[x], so the hyperoperation should be defined for elements of A.
  5. [Example 2.4] The displayed computation of 2^3∘3 in Example 2.4 does not match the hyperoperation with Φ={2,3}; please recalculate the set of products.
  6. [Example 2.6] In Example 2.6, the expression '2◦2 = {8,12}/∈P' should read '2◦2 = {8,12}⊈P'.
  7. [Theorem 2.20] In Theorem 2.20(i), η^{-1}(P2) is said to be a C-hyperideal of A2; it should be of A1. In part (ii), the text contains 'P1 is a is a' and says η(P1) is a C-hyperideal of A1; the latter should be A2.
  8. [Theorem 2.23] In the proof of Theorem 2.23, the notation '⊈S^{-1}Q' should be '⊈S^{-1}P'.
  9. [Conclusion and Section 4] The conclusion contains the typo 'it ,s radical', and Section 4 (Future work) contains a new Definition 4.1; including a formal definition in a future-work section is surprising and should be either moved to the main development or removed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new class is explicitly defined, and the structural theorems are derived from Definition 2.1 plus cited external facts; self-citations are definitional, not load-bearing.

full rationale

The paper introduces (u,v)-absorbing primary hyperideals by an explicit definition (Definition 2.1) that generalizes the author's earlier 1-absorbing primary hyperideal [5] and (u,v)-absorbing prime hyperideal [6] notions. No fitted parameter is renamed as a prediction, and no central theorem is assumed as its own input. The main structural results (Theorems 2.7, 2.12, 2.15, 2.18, 2.19) are proved from Definition 2.1 together with cited external facts such as Proposition 3.2 and Proposition 3.3 of [13] and Lemma 2.6 of [19]; these cited results are not restatements of the target claims. There are genuine proof gaps, most notably the unjustified inference in Theorem 2.7 from a∘b⊆rad(P) to a^n∘b^n⊆P, and the unproved equality P=rad(P∘M) in Proposition 2.8. However, these are correctness risks rather than circularity: they do not make a conclusion identical to an input or reduce a prediction to a fitted parameter. The self-citations [5,6] are used for motivation and terminology, and the hierarchy claim placing the new class between primary and (u,v)-absorbing prime hyperideals is not obtained by importing that hierarchy from the self-citations. Accordingly, the paper is substantially self-contained and the derivation chain is not circular; the low score reflects only the presence of minor non-load-bearing self-citations, not any reductive equivalence.

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

No empirical fitting occurs. The paper depends on standard hyperring axioms and on several unproved or lightly cited structural facts (the radical calculus for C-hyperideals, the equality rad(P∘M)=P, and Dedekind hyperdomain properties). These are ledger axioms rather than fitted parameters.

assumptions (5)
  • domain assumption Multiplicative hyperring axioms including weak distributivity and identity element
    The entire paper works in this structure from [15], stated in Section 1.
  • domain assumption C-hyperideal and strong C-hyperideal definitions from [13] and [14] allow radical to equal the set of elements with some power inside the ideal
    Theorems 2.7 and 2.15 rely on the C-hyperideal property to invert the inclusion in Proposition 3.2 of [13].
  • domain assumption Proposition 2.8 hypothesis that every hyperideal of A is a C-hyperideal
    This is an assumption inside Proposition 2.8, used to transfer subset conditions to radical conditions.
  • domain assumption Lemma 2.6 in [19] characterizing local hyperrings by the unit property
    Theorem 2.12 concludes locality from x+y being a unit; this rests on the cited lemma.
  • domain assumption Dedekind hyperdomain theory from [19], including invertibility of nonzero proper hyperideals
    Theorem 2.18 relies on it to conclude prime radicals are maximal.

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Pith. "Pith review of $(u,v)$-absorbing primary hyperideals in multiplicative hyperrings." pith.science (2026). https://pith.science/paper/QWOYOIKF

@misc{pith2026250602989,
  author       = {Pith},
  title        = {Pith review of: $(u,v)$-absorbing primary hyperideals in multiplicative hyperrings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QWOYOIKF}},
  note         = {Machine review of arXiv:2506.02989}
}
abstract

The present paper addresses the notion of $(u,v)$-absorbing primary hyperideals in commutative multiplicative hyperrings.

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