{"id":"f4b7c537-131b-49d5-8343-edaaeb278c2d","arxiv_id":"2506.24031","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For orders in number fields, a subring is associated if and only if it is both ideal-preserving and locally associated, yielding a quadratic-order classification of half-factorial domains.","lead":"This paper defines three ways a subring can relate to a larger ring (associated, ideal-preserving, locally associated) and proves that, for orders in number fields, being associated is equivalent to being both ideal-preserving and locally associated. It then uses this to classify half-factorial orders in quadratic fields and reports a computational count of such orders.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 4.12 and Corollary 4.13 are internally sound under the stated Dedekind/integral/nonzero-conductor assumptions; the residual risk is reproducibility of the computational survey and deferred external examples.","rationale":"The reader's weakest assumption points to the scope of Theorem 4.12 (Dedekind domain, integral extension, nonzero conductor), and I agree the theorem is only proved in that scope; however, I do not see a flaw inside that scope. The proof of Theorem 4.12 is coherent: the primary-conductor case reduces to finding, for α ∈ P\\I, an element αβ ∈ R outside the next power of P, which follows directly from ideal-preservation; Lemma 4.10 then supplies a unit multiplier. For non-primary conductors, Lemma 4.11 proves that each R + P_i^{a_i} inherits both properties, the primary case makes each such ring associated, and the CRT construction of β congruent to units modulo each P_i^{a_i} yields αβ ∈ ∩(R + P_i^{a_i}) = R, after which Lemma 4.10 applies again. I checked the potentially delicate assertion in Lemma 4.11 that relative primality to the smaller ideal A transfers to relative primality to the larger ideal J: if rR + A = R and A ⊆ J, then rR + J = R, so r is a unit modulo J; since J ⊆ R in this context, this is exactly the needed condition in R + J. The quadratic results also check out: Theorem 5.8 follows from the conductor-primeness lemma and inertness, Theorem 6.2 computes L(n,d) correctly by decomposing primes as inert/split/ramified, and Theorem 6.4 correctly reads off the imaginary-quadratic cases from L(n,d) and the fundamental unit. Consequently, I found no reason to move the verdict. The conditional status is retained because the computational survey and some examples depend on code and unpublished material that a reader cannot verify from the preprint alone. This is a reproducibility concern, not a mathematical objection to the central claim.","tokens_in":26548,"tokens_out":25363,"duration_ms":281755,"concrete_test":"Independently recompute the quadratic-order classification for a bounded range, e.g. all indices n ≤ 200 and squarefree d with |d| ≤ 200, using only the definitions in Section 2: test associated via R·U(T), ideal-preserving via Theorem 3.2(4) on primes dividing the conductor, and locally associated via |U(T)/U(R)| compared with |U(T/I)|/|U(R/I)|. Then confirm that the outputs agree with Theorems 5.8, 6.2, 6.4, and Corollary 6.5, and, separately, verify Example 4.9 directly from Theorem 3.1(3) and Theorem 3.4(4).","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify a load-bearing flaw in the central claim. Theorem 4.12's converse survives close reading: the primary-conductor case uses ideal-preservation to find β outside the next prime power, Lemma 4.10 converts relative primality to a unit multiple, and the CRT descent to non-primary conductors is supported by Lemma 4.11 and Theorem 4.6. Lemma 4.11's unit-lifting step is valid because if rR + I = R and J ⊇ I, then rR + J = R, so r is a unit modulo J in R + J. The quadratic classification (Theorems 5.8, 6.2, 6.4 and Corollary 6.5) is internally consistent with the definitions and standard splitting/inertia facts. The residual risk is not in the central argument: Proposition 6.7's count of 29,163 half-factorial orders is only supported by a linked GitHub repository, and Example 4.9 defers verification to the unpublished preprint [6]. These external dependencies justify the reader's CONDITIONAL verdict but do not shake the main equivalence theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26748,"tokens_out":25722,"duration_ms":271016,"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].","major_comments":[],"minor_comments":[{"comment":"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.","section":"§4, Example 4.9"},{"comment":"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.","section":"§6, Proposition 6.7"},{"comment":"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.","section":"§6, proof of Theorem 6.4"},{"comment":"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.","section":"§3, proof of Theorem 3.2"},{"comment":"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.","section":"Notation"}],"recommendation":"minor_revision","confidential_remarks":"The central mathematical content appears sound: the proof of Theorem 4.12 survives close reading, and the quadratic classification is consistent with the stated definitions. The main publication risk is the dependence of two claims on the external GitHub repository [16] and the unpublished preprint [6]. I would ask the authors to make the computational parts self-contained or clearly marked as computational, and to adjust the notation for integral closures. Neither issue blocks the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nSoft 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":27298,"tokens_out":2397,"would_cite":true,"duration_ms":26809,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A15","13F15","11R11","11R27","11R29"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that an order in a number field is 'associated' exactly when it is both ideal-preserving and locally associated.","keywords":["associated subring","ideal-preserving subring","locally associated subring","orders in number fields","conductor ideal","half-factorial domain","quadratic orders","elasticity"],"falsifier":"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.","tokens_in":26293,"feed_emoji":"🧮","tokens_out":10950,"duration_ms":104118,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two local conditions characterize 'associated' number-field orders","feed_subtitle":"In quadratic fields these checks reduce to inert primes and one unit-power comparison, locating half-factorial orders.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the motivating theorem that an order is half-factorial iff its integral closure is and certain index and unit conditions hold, framing the associated-subring relation.","marker":"[10]"},{"why":"Characterizes half-factorial orders in number fields via conductor factorization, extending Halter-Koch and giving the target application for associated orders.","marker":"[20]"},{"why":"Establishes the associated-order example and the elasticity results for radical conductor ideals and formal power series, used to motivate and test the new definitions.","marker":"[6]"},{"why":"Provides the exact sequence relating $U(R)$, $U(R/I)$, $U(\\bar R)$, and class groups that underlies the locally associated criterion.","marker":"[18]"},{"why":"Supplies the lemmas that ideals relatively prime to the conductor are invertible and every ideal class has such a representative, used in the local-association inheritance proof.","marker":"[5]"},{"why":"Gives the detailed proof that $|U(R)/U(\\bar R)|$ divides $|U(\\bar R/I)|/|U(R/I)|$, used to justify the divisibility $m\\mid L(n,d)$.","marker":"[14]"},{"why":"Provides the companion table and code classifying associated, ideal-preserving, locally associated, and half-factorial quadratic orders, the source of Proposition 6.7.","marker":"[16]"},{"why":"Original result that $\\mathbb{Z}[\\sqrt{-3}]$ is half-factorial and is the only non-maximal half-factorial order in non-real quadratic fields; Corollary 6.6 reproduces it through the new machinery.","marker":"[8]"}],"fun_headline_variants":["Two local conditions characterize associated orders","Associated orders: ideal-preserving plus locally associated","Associated iff ideal-preserving and locally associated","For number-field orders: associated = ideal-preserving + locally associated","Half-factorial orders: unique non-maximal in non-real quadratics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two local conditions characterize associated orders","Associated orders: ideal-preserving plus locally associated","Associated iff ideal-preserving and locally associated","For number-field orders: associated = ideal-preserving + locally associated","Half-factorial orders: unique non-maximal in non-real quadratics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002181,"raw_usage":{"total_tokens":8468,"prompt_tokens":984,"completion_tokens":7484,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":7407}},"tokens_in":600,"tokens_out":7484,"duration_ms":57973,"temperature":1.0,"reasoning_tokens":7407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:24:31.341869+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Factorization of algebraic integers","cited_arxiv_id":null,"evidence_quote":"Supplies the motivating theorem that an order is half-factorial iff its integral closure is and certain index and unit conditions hold, framing the associated-subring relation."},{"cited_title":"Elasticity in orders of an algebraic number field with radical conductor ideal and their rings of formal power series, 2025","cited_arxiv_id":null,"evidence_quote":"Establishes the associated-order example and the elasticity results for radical conductor ideals and formal power series, used to motivate and test the new definitions."},{"cited_title":"The conductor ideal of an order","cited_arxiv_id":null,"evidence_quote":"Supplies the lemmas that ideals relatively prime to the conductor are invertible and every ideal class has such a representative, used in the local-association inheritance proof."},{"cited_title":"The HFD property in orders of a number field","cited_arxiv_id":null,"evidence_quote":"Gives the detailed proof that $|U(R)/U(\\bar R)|$ divides $|U(\\bar R/I)|/|U(R/I)|$, used to justify the divisibility $m\\mid L(n,d)$."},{"cited_title":"Associated Quadratic Orders, March 2025","cited_arxiv_id":null,"evidence_quote":"Provides the companion table and code classifying associated, ideal-preserving, locally associated, and half-factorial quadratic orders, the source of Proposition 6.7."},{"cited_title":"Half-factorial domains in quadratic fields","cited_arxiv_id":null,"evidence_quote":"Original result that $\\mathbb{Z}[\\sqrt{-3}]$ is half-factorial and is the only non-maximal half-factorial order in non-real quadratic fields; Corollary 6.6 reproduces it through the new machinery."}],"review_version":1}