{"id":"7b39de16-c7c7-4d32-a878-ad917e16c61f","arxiv_id":"1908.05074","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines RR-proper categories and claims that their proper cones form a ring, but the proof is invalid and the example is false.","lead":"This math paper claims that every ring has an ideal category whose 'proper cones' form a ring. The supporting proof has serious gaps and the main example, Euclidean domains, appears to be wrong.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 relies on Lemma 3, which asserts every component of a proper cone is epimorphic; this is false in the RR-proper category L(F_2), so the proof of the ring-of-cones theorem does not go through.","rationale":"The reader's rejection is justified, but the most load-bearing failure is inside the proof of Theorem 3 rather than in the choice of examples. Lemma 3 is unproved and incompatible with Definition 3: properness only gives at least one epimorphic component. In the concrete RR-proper category L(F_2), the proper cone γ with γ(D)=id_D and γ(0)=0_{0,D} satisfies γ*=γ, yet γ(0) is not epi and its epimorphic component is id_0, not itself. Since Theorem 3 uses Lemma 3 to strip the epimorphic-component operation in the distributivity proof, that proof is invalid. The reader's Euclidean-domain concern is also sound: in Z, the interval between (0) and (6) contains (12), which has no relative complement, so the claimed RR-proper examples do not exist. Both defects independently prevent the main theorem from being established; the false lemma is independent of the example failure. I therefore keep the reader's REJECT verdict with no adjustment.","tokens_in":9305,"tokens_out":17703,"duration_ms":186981,"concrete_test":"Verify by hand: take D=F_2 and the category L(D) as in Section 4. Check that L(D) satisfies Definition 5 (objects 0 and D; Boolean lattice; all inclusions split; each object carries a proper cone). Define γ(D)=id_D and γ(0)=0_{0,D}, and check the cone condition j(0,D)∘γ(D)=γ(0). Compute d0=max{imγ(a)}=D and e=id_D, hence γ*=γ. Then compute the epimorphic component of γ(0): show that g∘γ(0)=h∘γ(0) for g=id_D and h=0_D although g≠h, or factor γ(0)=id_0∘j(0,D). This contradicts Lemma 3, and therefore the step 'by Lemma 3' in the proof of Theorem 3 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3, Lemma 3 (stated without proof) asserts that for the cone γ* of Lemma 2, the epimorphic component satisfies ((γ*)(a))^o = γ*(a) for every object a. Definition 3 only requires a proper cone to have at least one epimorphic component, so this does not follow. It is false in an RR-proper category. Let D=F_2 and take L(D) from Section 4. Its objects are 0 and D, hom(D,D)=D, hom(0,D) is the zero morphism, and the lattice {0,D} is Boolean, so L(D) satisfies the RR-conditions of Definition 5. Let γ be the cone with vertex D given by γ(D)=id_D and γ(0)=0_{0,D}. The image set is {0,D}, so d0=D and the unique retraction e:D→D is id_D; hence γ*=γ. Lemma 3 would imply that γ(0), the zero morphism 0→D, is its own epimorphic component. But this zero morphism is not epi: for g=id_D and h=0_D, g∘γ(0)=h∘γ(0) (both are the unique morphism 0→D), yet g≠h. Equivalently, its canonical factorization is id_0 followed by the inclusion 0→D, so its epimorphic component is id_0, not the zero morphism. Theorem 3 invokes Lemma 3 in both displayed distributivity chains, exactly to replace the epimorphic-component operation on (γ⊕δ)* by the cone itself. Since Lemma 3 is false, the distributivity proof is invalid and the central theorem is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a categorical framework for the ideal theory of rings. It defines proper categories and proper cones as generalizations of Nambooripad's normal categories and normal cones, then introduces 'RR-proper categories' by adding lattice-theoretic conditions on the object set. The main claimed result (Theorem 3) is that the set PC of all proper cones in an RR-proper category is a ring, with multiplication inherited from cone composition and addition defined via a 'sum cone' followed by an image retraction. Section 4 constructs the category L(R) of principal left ideals of a ring and claims that it is a preadditive proper category, that it is RR-proper under suitable lattice conditions, and that the ring PL(R) of proper cones can be formed and iterated.","tokens_in":9693,"tokens_out":13622,"duration_ms":137234,"significance":"If the main theorem were correct, it would provide a unified categorical description of ideal structure and a new construction of a ring from a category, extending prior work on regular semigroups and normal cones. The paper also contains a reasonable elementary verification that L(R) is a preadditive proper category. However, the central ring theorem is not established: Lemma 3 is false, Lemma 4 rests on an unjustified identification, the distributivity proof in Theorem 3 misapplies Lemma 4, and the claimed RR-property fails for Euclidean domains such as Z. No machine-checked proofs or reproducible code accompany the manuscript, and several proofs are only sketched. The conceptual direction is interesting, but the specific results as stated are not reliable.","major_comments":[{"comment":"Lemma 3 asserts that for the proper cone gamma* of Lemma 2, every component is its own epimorphic component, i.e. ((gamma*)(a))^o = gamma*(a) for all a. Definition 3 only guarantees that at least one component is epimorphic, so the lemma cannot follow from the definition alone. It is in fact false in the RR-proper category L(F_2) of Section 4. Take the cone gamma with vertex F_2 given by gamma(F_2)=id and gamma(0)=0_{0,F_2}. Then d0=F_2 and the unique retraction e is the identity, so gamma*=gamma and gamma is proper. But gamma(0) is the zero morphism 0 -> F_2, which is not epimorphic: in hom(F_2,F_2), id_{F_2} and 0_{F_2} are distinct yet compose with gamma(0) to the same zero arrow 0 -> F_2. The canonical factorization of gamma(0) is id_0 followed by the inclusion 0 -> F_2, so its epimorphic component is id_0, not gamma(0). Since Theorem 3 uses Lemma 3 in both distributivity chains to remove the operation (·)^o, the proof of the ring theorem is invalid.","section":"Section 3, Lemma 3"},{"comment":"The proof of Lemma 4 contains an unjustified equality involving the definition of beta*. In the displayed chain, the step from gamma(a)·[(beta(c_gamma))^o · e(c_beta,d0)] to gamma(a)·[beta*(c_gamma)]^o assumes that (beta(c_gamma))^o · e(c_beta,d0) equals (beta(c_gamma) · e(c_beta,d0))^o, with d0 the maximal image object of gamma·beta. But Lemma 2 defines beta* using a retraction e(c_beta,d0^beta), where d0^beta is the maximal image of beta itself; no argument shows that d0 coincides with d0^beta or that the retractions coincide. The subsequent cancellation of the operation (·)^o via Lemma 3 is therefore not licensed, and the equality (gamma·beta)* = gamma·beta* is unsupported.","section":"Section 3, Lemma 4"},{"comment":"Even taking Lemmas 3 and 4 at face value, the distributivity proof misapplies Lemma 4. In the first chain, the proof passes from [rho(a)·(gamma(c)j(c_gamma,d)+delta(c)j(c_delta,d))]^* to rho(a)·[(gamma(c)j(c_gamma,d)+delta(c)j(c_delta,d))]^* by invoking Lemma 4. Lemma 4 concerns the product of two proper cones, gamma·beta*; it says nothing about precomposing a single morphism rho(a) with a sum of morphisms. The displayed inference is not an instance of Lemma 4, and the parallel step in the second chain again relies on Lemma 3. The proof of distributivity, and hence of Theorem 3, does not go through.","section":"Section 3, Theorem 3"},{"comment":"The sentence 'It is easy to see that the ideal categories of Euclidean domains are RR−proper categories' is false. In L(Z), consider the interval of principal ideals from (0) to (6). The ideal (12) lies in this interval. For a principal ideal (n) to be a relative complement of (12) in this interval, one would need (12)∩(n)=(0) and (12)+(n)=(6). The first condition forces n=0, since gcd(12,n)=0 only for n=0, but then (12)+(0)=(12), not (6). Hence the principal ideals of a Euclidean domain need not form a relatively complemented lattice. Consequently the main theorem cannot be applied to L(Z), which is the most basic nontrivial example, and the claimed RR-properness of Euclidean ideal categories is not established.","section":"Section 4, RR-proper claim for Euclidean domains"},{"comment":"The proof of associativity of the semigroup PC is incomplete. The displayed calculation contains undefined notation, such as c_{αβ}, and unmatched parentheses; the step from the third to the fourth displayed line is not justified by any stated rule. Since the multiplication on PC is part of the ring structure later claimed in Theorem 3, the semigroup claim itself needs a full, correct proof.","section":"Section 3, Theorem 2"}],"minor_comments":[{"comment":"The heading 'Prelimanires' is a typo and should read 'Preliminaries'.","section":"Section 2 heading"},{"comment":"The notation gamma ⋆ f^o is used without a definition for arbitrary morphisms f; Lemma 1 only defines gamma ⋆ f for an epimorphism f. The proposition should specify how the construction is extended to the epimorphic component f^o.","section":"Proposition 1"},{"comment":"The proof has a bracketing error: in the second displayed line, the term δ(b)j(d,c∨d)] contains an extra closing bracket, and the displayed computation is difficult to follow as a result.","section":"Lemma 5 proof"},{"comment":"The notation ρa for the proper cone with vertex Ra is introduced informally; the earlier construction in Lemma 8 uses ρd with a fixed vertex Rd. The paper should define the cone ρa explicitly, including its components ρa(Rd) for all objects Rd.","section":"Section 4, final paragraph"},{"comment":"There are numerous grammatical and spelling errors (e.g. 'cotaining', 'isomorphism' in plural contexts) that should be corrected in a revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"I concur with the stress-test assessment: the central construction does not currently work, and the manuscript contains a false lemma together with an unsupported distributivity argument. The claimed application to Euclidean domains is also contradicted by the example of Z. A revision would require substantially new hypotheses and proofs, not merely local corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's main claim—that the proper cones of an RR-proper category form a ring—is not established. The key lemma used in the proof is false, and the one concrete family of examples offered, Euclidean domains, is also wrong. That said, the paper is not silly: it extends Lukose–Rajan's ring of normal cones to proper cones and introduces the RR-proper condition, which is a reasonable thing to try.\n\nWhat is genuinely new is the definition of RR-proper category and the observation that the principal ideal category of any ring is a preadditive proper category (Theorem 4). The definitions are written clearly enough that you can locate the failures precisely, which is a plus.\n\nThe failures are load-bearing. Lemma 3 asserts that every component of the proper cone γ* is its own epimorphic component. Definition 3 only requires one such component, so this is suspicious immediately. In the category L(F₂), with objects 0 and F₂, take the cone γ with γ(F₂)=id and γ(0)=0→F₂. It is proper, and γ*=γ because the maximal image is F₂. But γ(0) is the zero morphism, whose epimorphic component is id₀, not itself. Lemma 3 is false, and Theorem 3 uses it in both distributivity chains. Lemma 4's proof also contains an unjustified step: it swaps (γ(c))ᵒe with (γ(c)e)ᵒ, which is not generally true. Lemma 6's associativity proof is a formal manipulation that doesn't follow from the cone definitions.\n\nThe application to rings fails separately. Section 4 claims Euclidean domains are RR-proper because their principal ideal lattices are relatively complemented. In Z, the interval from (0) to (6) contains (12), but (12) has no relative complement: any D with (12)∩D=(0) must be (0), and then (12)+(0)≠(6). So the theorem never applies to the paper's own example.\n\nA specialist in categorical semigroup theory might find the RR-proper notion worth salvaging, but the ring-of-cones theorem does not hold as stated. I would send this to a knowledgeable referee rather than desk reject it—the flaws are pinpointable and the authors might repair some of them—but the report would be negative unless the lemmas are fixed.","headline":"The ring-of-cones theorem fails: Lemma 3 is false and the Euclidean domain example collapses, so the paper's central claims do not hold.","tokens_in":10199,"tokens_out":6635,"would_cite":false,"duration_ms":61367,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M10","20M12","18E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, for rings whose principal ideals satisfy two lattice conditions, the proper cones in the ideal category form a ring.","keywords":["Proper category","RR-proper category","Proper cones","Principal ideals","Preadditive category","Ring of cones","Euclidean domains"],"falsifier":"In $\\mathbb{Z}$, inspect the interval $[(0),(6)]$ of principal ideals. The ideal $(12)$ lies in this interval. A relative complement $D$ of $(12)$ would need $(12)\\cap D=(0)$ and $(12)+D=(6)$; the first condition forces $D=(0)$, but then $(12)+(0)=(12)\\neq(6)$. So no relative complement exists, and the principal ideals of the Euclidean domain $\\mathbb{Z}$ are not relatively complemented. This single interval check refutes the paper's claim that Euclidean domains satisfy the RR-condition and therefore blocks the theorem from applying to that example.","tokens_in":9088,"feed_emoji":"🧮","tokens_out":9232,"duration_ms":93469,"temperature":0.7,"pith_summary":"This paper tries to establish that the ideal structure of a unital ring can be studied through a category whose 'proper cones' are themselves ring elements. It shows that the principal left ideals of any ring, with right translations as morphisms, form a preadditive proper category, and dually for principal right ideals. It then introduces extra lattice conditions, called the RR-conditions, under which the proper cones form a ring with natural addition and multiplication. The payoff would be a uniform categorical way to build a new ring from a ring's ideal theory, and to iterate the construction. The paper claims Euclidean domains satisfy the extra conditions.","feed_headline":"Ideal categories turn rings into rings of cones","feed_subtitle":"A new construction gives the proper cones of an ideal category their own addition and multiplication.","key_machinery":"The load-bearing object is the RR-proper category together with the proper cones on it. A proper cone is a family of morphisms $\\gamma(c):c\\to d$ indexed by objects, compatible with inclusions, with at least one component an epimorphism; an RR-proper category is a preadditive proper category whose object poset is a relatively complemented lattice in which every bounded-above subset has a unique maximal element. The cone operations are defined by joining vertices and then retracting to the maximal image: $(\\gamma\\oplus\\delta)(a)=\\gamma(a)j(c_\\gamma,c_\\gamma\\vee c_\\delta)+\\delta(a)j(c_\\delta,c_\\gamma\\vee c_\\delta)$, with $\\gamma+\\delta=(\\gamma\\oplus\\delta)^*$, and multiplication $(\\gamma\\cdot\\eta)(a)=\\gamma(a)(\\eta(c_\\gamma))^o$. The RR-conditions make these operations well-defined and closed, and the preadditive structure makes composition distribute over addition.","core_discovery":"The central claim is that the categorical machinery of proper categories, previously used for semigroups, transfers to rings at the level of principal ideals. For a unital ring $R$, the objects are the principal left ideals $Ra$; a morphism $Ra\\to Rb$ is a right translation $x\\mapsto xs$ with $as\\in Rb$. This gives a category $L(R)$ that is proper and preadditive, and the dual category $R(R)$ of principal right ideals behaves the same way. If the poset of principal ideals further satisfies the RR-conditions, then $L(R)$ is an RR-proper category, and the paper's main theorem says the set of proper cones $P_C$ is a ring: multiplication is cone composition followed by the epimorphic component of the second cone at the first cone's vertex, and addition is formed by joining the two cone vertices, adding components into the join, and retracting to the maximum image. The theorem is stated for any RR-proper category $C$, so the ring of cones is not tied to a particular base ring.","pith_inferences":["My inference: the class of rings satisfying the RR-conditions is likely much narrower than the paper's Euclidean-domain claim suggests, so a useful next step would be to characterize rings whose principal ideals form relatively complemented lattices.","My inference: the iterated construction $R \\mapsto P_{L(R)} \\mapsto P_{L(P_{L(R)})}$ invites the question of when the cone ring is isomorphic to the original ring, or when it is commutative, semisimple, or Noetherian; the paper does not address these properties.","My inference: the construction could be tested computationally on finite rings by explicitly enumerating principal ideals and checking the complement and maximal-element conditions, which would show which small rings actually admit a nontrivial cone ring."],"forward_implications":["Every unital ring $R$ gives a preadditive proper category $L(R)$ of principal left ideals, and dually $R(R)$ of principal right ideals, so the cone construction has a natural home in any ring's ideal structure.","Whenever the RR-conditions hold, the proper cones form a ring whose additive identity is the cone at the zero ideal and whose additive inverse is given by negating each cone component.","The multiplication of proper cones is associative because cone composition preserves epimorphic components, and it distributes over the lattice-based addition because the underlying hom-sets are additive abelian groups.","The construction iterates: because the left and right ideal categories of the cone ring are again RR-proper categories under the same conditions, their proper cones again form rings at the next level."],"supporting_citations":[{"why":"Supplies the normal-cone ring construction and the lattice-and-retraction lemmas on images that the paper adapts from normal cones to proper cones.","marker":"[6]"},{"why":"Provides the normal category formalism of subobjects, split inclusions, normal factorizations, and the cone semigroup theorem on which proper categories are modelled.","marker":"[9]"},{"why":"Defines proper categories and the proper-cone semigroup; this paper extends that semigroup to a ring and instantiates it with the principal ideals of rings.","marker":"[11]"}],"fun_headline_variants":["Rings yield rings via ideal category cones","Ideal category cones form a new ring","Cones of ideal categories give a ring structure","Ring of cones from ideal categories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the principal ideals of the ring form a relatively complemented lattice in which every bounded-above subset has a unique maximal element; the paper relies on this to make the ideal category RR-proper, and without it the definition of cone addition, which retracts to a maximal image, can fail to be sound.","fun_headline_variants_meta":{"raw":{"variants":["Rings yield rings via ideal category cones","Ideal category cones form a new ring","Cones of ideal categories give a ring structure","Ring of cones from ideal categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000464,"raw_usage":{"total_tokens":2226,"prompt_tokens":763,"completion_tokens":1463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":379,"completion_tokens_details":{"reasoning_tokens":1409}},"tokens_in":379,"tokens_out":1463,"duration_ms":10633,"temperature":1.0,"reasoning_tokens":1409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:25:46.953146+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In $\\mathbb{Z}$, inspect the interval $[(0),(6)]$ of principal ideals. The ideal $(12)$ lies in this interval. A relative complement $D$ of $(12)$ would need $(12)\\cap D=(0)$ and $(12)+D=(6)$; the first condition forces $D=(0)$, but then $(12)+(0)=(12)\\neq(6)$. So no relative complement exists, and the principal ideals of the Euclidean domain $\\mathbb{Z}$ are not relatively complemented. This single interval check refutes the paper's claim that Euclidean domains satisfy the RR-condition and therefore blocks the theorem from applying to that example.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the normal-cone ring construction and the lattice-and-retraction lemmas on images that the paper adapts from normal cones to proper cones."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the normal category formalism of subobjects, split inclusions, normal factorizations, and the cone semigroup theorem on which proper categories are modelled."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines proper categories and the proper-cone semigroup; this paper extends that semigroup to a ring and instantiates it with the principal ideals of rings."}],"review_version":1}