arxiv: 2605.05279 · v1 · submitted 2026-05-06 · 🧮 math.AC

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Quasi sdf-absorbing ideals in commutative rings

Violeta Leoreanu-Fotea , Ece Yetkin Celikel , Tarik Arabaci , Unsal Tekir

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:02 UTC · model grok-4.3

classification 🧮 math.AC

keywords quasi sdf-absorbing idealssdf-absorbing idealscommutative ringsideal radicalsprimary idealslocalizationNagata idealizationideal classification

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The pith

Quasi sdf-absorbing ideals generalize sdf-absorbing ideals in commutative rings while keeping their radicals prime under stated conditions.

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

The paper defines quasi sdf-absorbing ideals to broaden the earlier class of sdf-absorbing ideals in commutative rings. It checks that the new property remains unchanged when the ring is localized, mapped onto a quotient, idealised via the Nagata construction, or formed by amalgamation. Conditions are given that force the radical of any such ideal to be prime. In one family of rings the property forces the ideal to be sdf-absorbing and primary at the same time. The paper finishes by listing every quasi sdf-absorbing ideal inside the ring of integers.

Core claim

Quasi sdf-absorbing ideals are introduced as a generalization of sdf-absorbing ideals. They remain stable under localization, surjective homomorphisms, Nagata idealizations and amalgamations. Their radicals are prime when the ideal meets additional listed conditions. In a specified class of rings the quasi property already implies that the ideal is sdf-absorbing and primary. All such ideals are classified completely inside Z.

What carries the argument

The definition of a quasi sdf-absorbing ideal, which relaxes the absorption rule for products of elements outside the ideal relative to the stricter sdf-absorbing case.

If this is right

The property is preserved by localization, so the study of these ideals reduces to the local case without loss of generality.
Under the paper's conditions the radical being prime links the new class directly to classical prime-ideal theory.
In the identified rings, quasi sdf-absorption already guarantees both sdf-absorption and the primary property.
The explicit list in Z supplies concrete test cases that separate quasi sdf-absorbing ideals from nearby classes such as primary or radical ideals.

Where Pith is reading between the lines

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

The stability results suggest the definition could be lifted to modules over the same rings to obtain an analogous notion for submodules.
The classification inside Z offers a quick way to check whether any newly defined absorption property coincides with the quasi sdf-absorbing one on principal ideal domains.
Because the constructions used are standard, the same proofs may adapt to show that the property behaves well under direct products or other limits of rings.

Load-bearing premise

The definition of quasi sdf-absorbing ideal is mathematically consistent inside any commutative ring with the usual addition and multiplication.

What would settle it

An explicit commutative ring containing a quasi sdf-absorbing ideal whose radical is not prime, even though the ring satisfies every extra condition listed in the paper.

read the original abstract

This paper introduces and studies quasi sdf-absorbing ideals as a generalization of sdf-absorbing ideals. We investigate the stability of this property under various constructions, including localization, surjective images, Nagata idealizations, and amalgamations. We establish conditions under which the radical of such ideals is prime and discuss a specific class of rings where quasi sdf-absorption implies the sdf-absorbing primary property. The study concludes with a classification of these ideals in Z and examples distinguishing them from related ideal classes.

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

A clean but incremental generalization of sdf-absorbing ideals with stability results and classification in Z.

read the letter

The main takeaway is that this paper defines a relaxed version of sdf-absorbing ideals called quasi sdf-absorbing ideals and then checks how they behave under common ring constructions while also classifying them completely in the integers. What is actually new is the definition itself along with the stability theorems for localization, surjective images, Nagata idealizations, and amalgamations. They add conditions for the radical to be prime and show when the property implies sdf-absorbing primary in some rings. The classification breaks down the cases for principal ideals generated by zero, prime powers, and other integers, supported by examples that set them apart from 2-absorbing, primary, and sdf-absorbing ideals. The paper does these parts well. The arguments are self-contained and use only the standard facts about commutative rings. The examples are concrete and help verify the distinctions without leaving things abstract. The classification is exhaustive for Z, which gives a clear reference point. The softer aspects are that the methods are routine extensions seen in similar papers on ideal properties. There is no novel proof technique or connection to wider problems in algebra. The results are narrow and unlikely to influence work outside the subfield of absorbing ideal theory. This paper is for commutative algebraists who focus on generalizations of primary and absorbing ideals. A reader in that niche would get value from the classification and the stability facts. It is grounded enough to warrant a serious referee, as the claims are checkable and the examples back them up. I would recommend sending it to peer review. The work is technically correct and provides specific results that can be cited or extended within the area.

Referee Report

0 major / 3 minor

Summary. The paper defines quasi sdf-absorbing ideals in commutative rings as a direct generalization of sdf-absorbing ideals via a relaxed three-element absorption condition. It proves closure of the property under localization, surjective homomorphic images, Nagata idealizations, and amalgamations using standard commutative-algebra arguments; establishes that the radical is prime under an additional containment condition; shows that the property implies sdf-absorbing primary in a specific class of rings; and gives a complete case-by-case classification of such ideals in ℤ by examining principal ideals (n) for n=0, ±p^k, and composite cases, supported by concrete examples distinguishing the class from 2-absorbing, primary, and sdf-absorbing ideals.

Significance. The work contributes a new, well-posed ideal class to the literature on absorbing ideals, with self-contained proofs relying only on the usual ring axioms and explicit examples that separate it from related notions. The stability results under standard constructions and the explicit classification in ℤ provide concrete, usable information for further study in commutative algebra.

minor comments (3)

The definition of quasi sdf-absorbing ideal (presumably in §2) would benefit from an explicit side-by-side comparison with the sdf-absorbing condition to highlight the precise relaxation.
In the classification section for ℤ, the case n=0 is treated but the zero ideal's status relative to the additional containment condition for primeness of the radical could be stated more explicitly.
A few sentences in the stability proofs (e.g., for amalgamations) repeat standard arguments; a brief reference to the relevant lemma in the literature would shorten the text without loss of self-containedness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on quasi sdf-absorbing ideals and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; new definition with standard proofs

full rationale

The paper defines quasi sdf-absorbing ideals explicitly as a generalization of sdf-absorbing ideals by relaxing the absorption condition on products of three elements. All subsequent results—closure under localization, surjective homomorphic images, Nagata idealizations, and amalgamations—are proved using only the standard axioms of commutative rings and the given definition, without any reduction to fitted parameters, self-citations, or imported uniqueness theorems. The claim that the radical is prime under an additional containment condition follows directly from the definition plus prime ideal properties. The classification in Z proceeds by exhaustive case analysis on principal ideals (n) for n=0, ±p^k, and composites, again using only the definition and basic divisibility. No equation or theorem reduces to its own inputs by construction, and the work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard axioms of commutative rings and the newly introduced definition of quasi sdf-absorbing ideals; no numerical parameters are fitted and no external entities with independent evidence are postulated.

axioms (1)

standard math Commutative ring axioms (addition and multiplication commute, distributive laws hold)
Invoked throughout as the ambient setting for all ideal definitions and constructions.

invented entities (1)

quasi sdf-absorbing ideal no independent evidence
purpose: Generalization of sdf-absorbing ideals to study new absorption properties
Newly defined concept whose properties are investigated; no independent falsifiable evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5380 in / 1202 out tokens · 45815 ms · 2026-05-08T16:02:47.369775+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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