An analogue of Rogers' theorem on sieving in commutative rings

Petr Kucheriaviy

arxiv: 2603.14080 · v2 · submitted 2026-03-14 · 🧮 math.AC · math.NT

An analogue of Rogers' theorem on sieving in commutative rings

Petr Kucheriaviy This is my paper

Pith reviewed 2026-05-15 11:19 UTC · model grok-4.3

classification 🧮 math.AC math.NT

keywords Rogers theoremsievingDedekind domaincommutative ringlocal ringidealsordersfinite rings

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The pith

An analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain, and for a finite commutative ring if and only if the ring is a direct product of local rings with linearly ordered ideals.

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

The paper gives if-and-only-if characterizations for when a sieving property analogous to Rogers' theorem holds inside commutative rings. For an order, the property is equivalent to the order being a Dedekind domain. For a finite commutative ring, the property holds exactly when the ring factors as a direct product of local rings whose ideals form a linear chain under inclusion. A sympathetic reader would care because the result translates a concrete counting or distribution condition into exact algebraic structure, showing which rings inherit the sieving behavior from the integers.

Core claim

We prove that an analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain. We also prove that it holds for a finite commutative ring if and only if the ring is a direct product of local rings with linearly ordered ideals.

What carries the argument

The analogue of Rogers' theorem on sieving, a counting or distribution condition on elements of the ring that the paper equates to the ring's ideal structure.

If this is right

An order satisfies the sieving analogue only when it is a Dedekind domain.
Every Dedekind domain satisfies the sieving analogue.
A finite commutative ring satisfies the analogue only when it decomposes as a direct product of local rings whose ideals are linearly ordered.
In such finite rings the ideal lattice of each local factor is a chain.

Where Pith is reading between the lines

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

The result supplies a sieving test that could be used in practice to recognize whether a given order is Dedekind.
The linear ordering of ideals may simplify explicit sieving calculations inside those finite rings.
The same style of characterization might be attempted for other classes of rings, such as orders in function fields.
The sieving property appears to detect global regularity of the ring rather than local properties alone.

Load-bearing premise

The precise statement of the analogue of Rogers' theorem on sieving is fixed in advance and the proofs use only standard facts about Dedekind domains and local rings with chain ideals.

What would settle it

Exhibit an order that is not a Dedekind domain yet satisfies the sieving condition, or a Dedekind domain that fails the condition; or exhibit a finite commutative ring that is not a direct product of local rings with linearly ordered ideals yet satisfies the condition.

read the original abstract

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

The paper gives two if-and-only-if characterizations tying an analogue of Rogers' sieving theorem to Dedekind domains and to finite rings that are products of local rings with chained ideals.

read the letter

The main point is that the author shows the sieving analogue holds for an order exactly when the order is Dedekind, and holds for a finite commutative ring exactly when the ring is a direct product of local rings whose ideals form a chain. These equivalences are presented as new and are not flagged as already known in the cited work. The paper does a reasonable job of connecting the sieving idea to standard ring-theoretic properties like unique ideal factorization and the structure of Artinian rings with linear ideal chains, which could give a compact way to recognize these classes through a single property. If the proofs are direct and avoid extra assumptions, the characterizations could be handy for people working on ring classifications in algebraic number theory. The soft spots are mostly around presentation. The abstract does not spell out the precise sieving condition being used, so it is difficult to judge how much is a genuine extension versus a rephrasing of known facts about Dedekind domains and principal ideal rings. Without the full proofs it is impossible to check for gaps or to see whether the arguments rely only on standard tools or introduce any subtle restrictions. That said, nothing in the claims looks internally inconsistent or circular on the surface. This paper is for specialists in commutative algebra who already know Rogers' theorem and care about ideal-theoretic characterizations of rings. A reader who wants precise equivalences for classifying orders or finite rings would get some value, though the impact stays inside the subfield. It deserves a serious referee because the statements are specific enough to be checked or refuted once the definition and proofs are on the table. Recommendation: send it to peer review so the details can be examined properly.

Referee Report

0 major / 2 minor

Summary. The paper proves that an analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain. It also proves that the analogue holds for a finite commutative ring if and only if the ring is a direct product of local rings with linearly ordered ideals.

Significance. If the proofs are correct, the results give clean algebraic characterizations of Dedekind domains and of finite rings whose ideals form chains, linking a sieving property to standard classes in commutative algebra. This may be useful for studying ideal lattices and factorization without introducing new parameters or ad-hoc constructions.

minor comments (2)

The abstract states the two iff claims clearly but does not include the precise formulation of the sieving analogue; this definition should appear explicitly in §1 or §2 so that the reader can verify the statements without external lookup.
Notation for the sieving condition and for the orders/rings under consideration should be introduced once and used consistently throughout the proofs of the two main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the two main theorems: the sieving analogue characterizes Dedekind domains among orders, and characterizes finite rings that are products of local rings with linearly ordered ideals. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states two iff characterizations linking an analogue of Rogers' sieving theorem to Dedekind domains for orders and to direct products of local rings with linearly ordered ideals for finite commutative rings. These rest on standard external facts from commutative algebra (unique ideal factorization in Dedekind domains, Artinian structure with chained ideals) without any quoted equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the claims to their own inputs. The derivation chain is therefore self-contained against independent ring-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard axioms of commutative ring theory and the prior definition of Rogers' theorem; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard axioms and definitions of commutative rings, orders, Dedekind domains, and local rings
Invoked throughout any proof of the stated equivalences in commutative algebra.

pith-pipeline@v0.9.0 · 5325 in / 1230 out tokens · 64984 ms · 2026-05-15T11:19:25.594845+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that an analogue of Rogers’ theorem on sieving holds for an order if and only if the order is a Dedekind domain.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

any finite set of ideals in R satisfies (Rg) iff R is a direct product of local rings with linearly ordered ideals

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.