Multiplicative Relationships of Subrings and their Applications to Factorization

Grant Moles

Multiplicative Relationships of Subrings and their Applications to Factorization

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that an order in a number field is 'associated' exactly when it is both ideal-preserving and locally associated.

desk verdict A clean structural equivalence (associated = ideal-preserving + locally associated) for orders, with a complete imaginary-quadratic classification; the main math looks solid, but reproducibility of the computational survey hangs on an unverified link. read the letter →

arxiv 2506.24031 v2 pith:RWVWSP7V submitted 2025-06-30 math.AC

Grant Moles This is my paper · ORCID

classification math.AC MSC 13A1513F1511R1111R2711R29

keywords associatedsubringideal-preservinglocallyordersinnumberfieldsconductoridealhalf-factorialdomainquadraticelasticity

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 asks when a subring $R$ of a better-understood ring $T$ shares enough of $T$'s multiplicative structure that every element of $T$ differs from an element of $R$ only by a unit. It defines three such relationships—'associated' ($T=R\cdot U(T)$), 'ideal-preserving' (intersection with $R$ preserves containment of ideals), and 'locally associated' (a unit-group condition modulo the conductor ideal $I=(R:T)$)—and proves that for a Dedekind domain $T$ with $I\neq 0$ and $T$ integral over $R$, the first is exactly the conjunction of the other two. For orders in number fields this says an associated order is precisely one that is both ideal-preserving and locally associated, turning a global question about factorizations into two finitely checkable conditions. These conditions are explicit enough in quadratic fields to classify the half-factorial orders there, including the recovery of the classical fact that $\mathbb{Z}[\sqrt{-3}]$ is the only non-maximal half-factorial order in an imaginary quadratic field.

What carries the argument

The central objects are three subring relationships defined relative to an ambient ring $T$: an associated subring satisfies $T=R\cdot U(T)$; an ideal-preserving subring has $R\cap J_1\not\subseteq J_2$ whenever $J_1\not\subseteq J_2$ are $T$-ideals; and a locally associated subring satisfies $U(T)/U(R)\cong U(T/I)/U(R/I)$ for the conductor ideal $I=(R:T)$. The proof machinery for the main equivalence is conductor factorization into prime powers, Chinese Remainder Theorem constructions of elements $\beta$ congruent to units modulo each prime power, and Lemma 4.10, which converts an $R$-multiple by an element relatively prime to $I$ into an $R$-multiple by a genuine unit. In the quadratic-order application the load-bearing identity is the function $L(n,d)$, defined multiplicatively by the Legendre-symbol decomposition of primes, which computes $|U(T/I)|/|U(R/I)|$ where $T$ is the full ring of algebraic integers; comparing the least power of the fundamental unit lying in $R$ with $L(n,d)$ decides local association.

What would settle it

To test the central equivalence, compute an order $R$ in a number field with conductor $I\neq 0$ that satisfies both local conditions but fails to satisfy $R = R\cdot U(T)$; the paper's own classification code for quadratic orders can be extended past index 10000 or to cubic fields to search for such a case. For the quadratic corollary specifically, the falsifying observation would be a non-maximal half-factorial order in an imaginary quadratic field other than $\mathbb{Z}[\sqrt{-3}]$: the paper predicts none exists, so any such order would refute Corollary 6.6.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the equivalence in Theorem 4.12: if $T$ is a Dedekind domain, $R\subseteq T$ is a subring with identity, the conductor ideal $I=(R:T)$ is nonzero, and $T$ is integral over $R$, then $R$ is an associated subring of $T$ if and only if $R$ is both ideal-preserving and locally associated. The proof factors the conductor into prime powers $I=P_1^{a_1}\cdots P_k^{a_k}$, uses the Chinese Remainder Theorem to construct elements congruent to units modulo each prime power, and then applies Lemma 4.10 to promote the resulting local unit multiples to a genuine associate in $R$. For orders in number fields (Corollary 4.13), this says 'associated order' and 'ideal-preserving plus locally associated order' are synonymous. In quadratic fields the paper then gives explicit tests: an ideal-preserving order of index $n$ is one in which every rational prime divisor of $n$ is inert in $K=\mathbb{Q}(\sqrt{d})$, and a locally associated order is one in which the least power of the fundamental unit lying in $R$ equals the arithmetic function $L(n,d)$ defined in Section 6. The paper closes with the consequence that, among non-real quadratic fields, the index-2 order in $\mathbb{Q}(\sqrt{-3})$, namely $\mathbb{Z}[\sqrt{-3}]$, is the unique non-maximal half-factorial order.

Load-bearing premise

The load-bearing premise is that $T$ is a Dedekind domain, $R$ contains the identity of $T$, the conductor ideal (the elements of $T$ taking $T$ into $R$) is nonzero, and $T$ is integral over $R$; the equivalence is proved only under these hypotheses, and the paper says the general case is still under investigation.

Editorial extensions

If this is right

An order in a number field is an associated order exactly when it is both ideal-preserving and locally associated, so checking the association condition reduces to two finite computations.
In a quadratic number field $\mathbb{Q}(\sqrt{d})$, an order of index $n$ is ideal-preserving exactly when every rational prime divisor of $n$ is inert in the field; this is decidable by Legendre symbols.
In a non-real quadratic field, the only associated order of index $n>1$ is the index-2 order in $\mathbb{Q}(\sqrt{-3})$; consequently $\mathbb{Z}[\sqrt{-3}]$ is the only non-maximal half-factorial order in any imaginary quadratic field.
For real quadratic fields, a computer search of orders of index $n\le 10000$ with $d<1000$ finds exactly 29,163 half-factorial orders, providing a data set for further pattern analysis.
If $R$ is locally associated in $T$, then both $R[x]\subseteq T[x]$ and $R[[x]]\subseteq T[[x]]$ inherit the locally associated property, extending the reach of the unit-group comparison to polynomial and power-series rings.

Reading between the lines

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

Pith inference: The same equivalence may fail outside the Dedekind-and-integral setting; the paper itself notes the general case is still under investigation, so a natural next step is to test whether ideal-preserving plus locally associated remains sufficient in, say, Krull domains or non-integral extensions.
Pith inference: Since ideal-preserving orders in quadratic fields are already characterized by inert primes, the search for associated orders is really a search for locally associated orders; the paper's table suggests patterns in the minimal unit power $m$ that could be worth formulating as a conjecture relating $m$ to continued fractions or class groups of real quadratic fields.
Pith inference: The computational classification is extensible: running the same unit-power comparison for orders of index beyond 10000 or in higher-degree number fields would test how stable the 29,163 count and the quadratic patterns are, though such searches quickly become expensive as the index grows.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: Theorem 4.12 and Corollary 4.13 are the real content and they are sound. The paper earns its keep by showing that two independently motivated subring properties together characterize associated subrings in the Dedekind-integral setting, and the quadratic corollary is a complete, clean answer for imaginary quadratic orders.

What is new: the three definitions were mostly drawn from the author's dissertation and from [6], but the systematic comparison, the equivalent characterizations, and especially the main equivalence theorem are new. The paper also derives an arithmetic function L(n,d) from first principles about unit groups and uses it to give an exact classification for non-real quadratic orders. That is a genuine contribution.

The proofs are by hand and mostly transparent. The examples involving Z[5√2], Z[2√2], and Q[x] are checked using the paper's own characterizations rather than asserted. Theorem 5.8, reducing ideal-preserving orders in quadratic fields to inert rational primes, is a nice and useful criterion. The paper also honestly flags that the general case of Theorem 4.12 is still under investigation, which is the right amount of ambition.

Soft spots, in proportion: First, the computational count of 29,163 half-factorial orders in Proposition 6.7 is supported only by a GitHub link and code not included in the preprint. That is a reproducibility gap, not a mathematical error, but an editor should ask for the code and table to be archived or for the claim to be softened. Second, Example 4.9 and Theorem 4.3 defer to an unpublished preprint [6]; a standalone paper should either prove those or point to a publicly available version. Third, some definitions and examples come from the author's prior dissertation, so a reader without access to that work has to take several things on faith. None of these undermines the central theorem.

The citation pattern is mostly appropriate for a small field, but the heavy reliance on the unpublished [6] is a real limitation for verification. That said, the central equivalence survives close reading, the imaginary quadratic classification is exact, and the computational survey is a reasonable exploratory addition if the artifact becomes available.

Who is this for: people working on factorization in orders, elasticity, and subring properties in commutative algebra. It deserves a serious referee, and the referee should focus on the external dependencies rather than the main proof. Send it to review.

Referee Report

0 major / 5 minor

Summary. The paper introduces three ways in which a subring R of a commutative ring T can be compared multiplicatively to T: being an associated subring, an ideal-preserving subring, and a locally associated subring. It develops several equivalent characterizations of these notions, with simplifying versions in the Dedekind-domain and number-field settings. The central structural result, Theorem 4.12, states that when T is a Dedekind domain, R contains 1, the conductor I=(R:T) is nonzero, and T is integral over R, then R is an associated subring of T if and only if R is both ideal-preserving and locally associated; Corollary 4.13 specializes this to orders in number fields. The remainder of the paper applies these notions to quadratic orders: Theorem 5.8 characterizes ideal-preserving quadratic orders by the inertness of the rational primes dividing the index, an arithmetic function L(n,d) is defined and shown to compute the relevant unit-group ratio, and Theorem 6.4 together with Corollary 6.5 classifies locally associated and associated orders in imaginary quadratic fields. A computational survey of half-factorial orders in real quadratic fields is also reported.

Significance. If the results are correct, the paper provides a clean and non-obvious bridge between three individually checkable subring properties and the more global associatedness condition, with direct consequences for factorization theory. The central equivalence is derived from the definitions and standard algebraic number theory with no fitted parameters, and several nontrivial examples (Z[5√2], Z[2√2], Q[x]) are checked using the paper's own characterizations. The function L(n,d) and the classifications in Section 6 are explicit and potentially useful for further searches. I found no load-bearing technical error in Theorem 4.12 or in the quadratic classification. The main weaknesses are reproducibility-related rather than mathematical: Example 4.9 and Proposition 6.7 rely on an external GitHub repository, and parts of the exposition rely on the unpublished preprint [6].

minor comments (5)

[§4, Example 4.9] The assertion that Z[3√2] and Z[11√2] are associated while Z[33√2] is not locally associated is deferred to the GitHub table [16] rather than proved or reproduced in the text. Since this example illustrates the failure of intersection inheritance, please include the computation or state clearly that the verification is computational.
[§6, Proposition 6.7] The exact count of 29,163 half-factorial orders rests entirely on code and a table located at [16]; no details of the search, the verification procedure, or the precise set of pairs (n,d) are included. Please include the code and table as supplementary material, or explicitly label this as a computational observation rather than a theorem.
[§6, proof of Theorem 6.4] The claims that L(n,d)=1, 2, or 3 only in the listed cases are left as 'by inspection'; a short argument using the multiplicativity of L and the positivity of p-(d/p) for odd primes would make the classification fully verifiable.
[§3, proof of Theorem 3.2] In the proof that Condition 4 implies Condition 3, the case I=0 is dismissed with 'then T=R=I', which is not generally true (for example, Z⊆Q has conductor 0 and R≠T). Since Conditions 3 and 4 coincide for I=0 under the stated convention, a one-line separate argument would be cleaner and avoid the erroneous statement.
[Notation] The integral closure is repeatedly typeset identically to the subring R (for instance in Theorem 1.1, Theorem 5.3, and Example 5.10); using \overline{R} consistently would remove a real source of confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central equivalence is derived from definitions and standard facts, with only minor non-load-bearing self-citations.

full rationale

The paper's central result, Theorem 4.12 / Corollary 4.13, is not circular. It proves the equivalence between the definitional property "associated" and the conjunction of the independently defined properties "ideal-preserving" and "locally associated" using the factorization of the conductor, the Chinese Remainder Theorem, and the unit-lifting lemmas (Lemmas 4.10 and 4.11). The primary-conductor case and the descent to non-primary conductors do not presuppose the conclusion. The quadratic analysis (Theorems 5.8, 6.2, 6.4 and Corollary 6.5) computes unit-group quotients from standard prime-splitting facts and Dirichlet's unit theorem, with no fitted parameters or prediction-by-construction. Self-citations do appear: the definitions were introduced in the author's prior work [15] and [6], the detailed proof of Lemma 3.3 is deferred to the dissertation [14], Theorem 4.3 is quoted from [6], and Proposition 6.7 relies on the author's computational table at [16]. None of these is load-bearing for the main theorem; Lemma 3.3 originates from Neukirch [18], and the cited table is an external artifact rather than an input to the derivation. The residual risk is reproducibility of the computational count in Proposition 6.7 and deferred verification of Example 4.9 in [6], which are correctness or reproducibility concerns, not circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 2 invented entities

The paper's central claim rests on standard algebraic number theory (Dedekind domains, conductors, unit groups, class groups) and on the cited Lemma 3.3. The new quantities introduced (L(n,d) and the three subring properties) are definitions rather than fitted parameters, and the main equivalence is proven rather than assumed.

free parameters (1)

No free parameters
The theorems are proven from definitions with no fitted constants. The computational claim depends on the choice of search bounds n≤10,000 and d<1,000, but these are not fitting parameters; they are the domain of the survey.

assumptions (3)

domain assumption T is a Dedekind domain and T is integral over R, with nonzero conductor ideal (R:T).
Theorem 4.12 is stated only under these hypotheses, and the proof uses unique factorization of ideals into primes and the Chinese Remainder Theorem. The paper explicitly notes the general question remains open.
standard math Lemma 3.3, the exact sequence relating class groups and unit groups of an order in a number field, is cited from [18] with proof reference [14].
This lemma underlies the locally associated characterizations (Theorem 3.4) and the divisibility m|L(n,d) in Corollary 6.3. It is a standard result in algebraic number theory, though the paper relies on it without proof.
standard math Dirichlet's Unit Theorem and the standard decomposition of primes in quadratic fields (inert, split, ramified) are used.
These are classical results invoked in Section 6 to compute unit groups and conductor norms, e.g., U(OK) = {±u^k} and the Legendre symbol criteria.

invented entities (2)

The arithmetic function L(n,d) independent evidence
purpose: Encodes the ratio |U(R/I)| / |U(R/I)| for quadratic orders, giving a finite criterion for the locally associated property.
L(n,d) is a new defined object, but it is explicitly tied to computable invariants (Euler's totient, Legendre symbols, and unit group orders) and yields a falsifiable prediction for each quadratic order: local association holds iff m = L(n,d), where m is the minimal power of the fundamental unit in the order.
The three subring notions: associated, ideal-preserving, locally associated
purpose: Organizing definitions for comparing multiplicative structures of a subring and its overring.
These are new definitions (though the paper says they originally appeared in [15] and [6]), but they are not physical entities. Their usefulness is validated by the theorems and applications, not by external evidence.

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Cite this review

Pith. "Pith review of Multiplicative Relationships of Subrings and their Applications to Factorization." pith.science (2026). https://pith.science/paper/RWVWSP7V

@misc{pith2026250624031,
  author       = {Pith},
  title        = {Pith review of: Multiplicative Relationships of Subrings and their Applications to Factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWVWSP7V}},
  note         = {Machine review of arXiv:2506.24031}
}

abstract

When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being associated, being ideal-preserving, or being locally associated. We then explore how these properties of a subring might be leveraged to give information about $R$, including applications to the field of factorization. Of particular interest is the result that an order in a number field is associated if and only if it is both ideal-preserving and locally associated. We conclude with a discussion of how these properties are realized in the case of orders in a number field and how such orders might be found.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Elasticity of Orders from the $S$-relative Davenport Constant: an Arithmetic Application of a Number-Theoretic Investigation
math.AC 2026-05 unverdicted novelty 7.0 of 10

Exact elasticity formulas for orders with prime or primary conductor are proven via a new S-relative Davenport constant, with a partial resolution of the intermediate-order elasticity conjecture.
(Locally) Associated Subrings in Polynomial and Power Series Extensions
math.AC 2026-08 accept novelty 6.0 of 10

Full necessary-and-sufficient conditions are given for coefficient-varying polynomial and power series rings to be (locally) associated subrings of larger such rings, with consequences for half-factorial power series ...
On $q$-real and $q$-complex numbers
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For every real x > 1 the q-real series [x]_q converges in the disk |q| < 3−2√2 to a nonvanishing holomorphic function, partially proving the radius-convergence conjecture.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [6]

Elasticity in orders of an algebraic number field with radical conductor ideal and their rings of formal power series, 2025

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work page 2025

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work page 2013

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Jason Boynton, Jim Coykendall, Grant Moles, and Chelsey Morrow. Overrings of half-factorial orders,

work page

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work page 1960

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Hyun Seung Choi. Class group and factorization in orders of a PID. Journal of Number Theory, 265:226– 269, 2024

work page 2024

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The conductor ideal of an order

Keith Conrad. The conductor ideal of an order. Expository Paper, 2019. https://kconrad.math.uconn. edu/blurbs/gradnumthy/conductor.pdf

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Jim Coykendall. A characterization of polynomial rings with the half-factorial property. In Factorization in integral domains (Iowa City, IA, 1996), volume 189 of Lecture Notes in Pure and Appl. Math., pages 291–294. Dekker, New York, 1997

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