Ideal categories of rings and ring of cones

P.G. Romeo; Sreejamol P.R.

REVIEW 5 major objections 5 minor 12 references

Ideal categories of rings and ring of cones

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for rings whose principal ideals satisfy two lattice conditions, the proper cones in the ideal category form a ring.

desk verdict 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. read the letter →

arxiv 1908.05074 v1 pith:VQJJG4XJ submitted 2019-08-14 math.CT

Sreejamol P.R. , P.G. Romeo This is my paper · ORCID

classification math.CT MSC 20M1020M1218E05

keywords PropercategoryRR-properconesPrincipalidealsPreadditiveRingofEuclideandomains

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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 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.

What 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.

The 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.

The 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.

A 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.

Referee Report

5 major / 5 minor

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.

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 (5)

[Section 3, Lemma 3] 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 3, Lemma 4] 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 3, Theorem 3] 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 4, RR-proper claim for Euclidean domains] 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 3, Theorem 2] 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.

minor comments (5)

[Section 2 heading] The heading 'Prelimanires' is a typo and should read 'Preliminaries'.
[Proposition 1] 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.
[Lemma 5 proof] 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 4, final paragraph] 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.
[Throughout] There are numerous grammatical and spelling errors (e.g. 'cotaining', 'isomorphism' in plural contexts) that should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the ring-of-cones theorem is a conditional proof attempt; its main risks are an unproved Lemma 3 and a false Euclidean-domain claim, not input-output circularity.

full rationale

The derivation chain is definitional rather than predictive: Definition 5 introduces RR-proper categories so that cones can be added by (γ⊕δ)* and multiplied by γ·η = γ⋆η(cγ)^o, and Theorem 3 attempts to prove distributivity from those definitions. No data are fitted and no quantity is renamed as a prediction. The self-citation [11] in the introduction (proper cones form a semigroup) is not load-bearing because Theorem 2 re-proves the semigroup structure from Definitions 3 and 4. Proposition 3 and Lemma 5 are attributed to [6], but proofs are reproduced in the text, so the citation is not the only support. The serious issues are correctness gaps, not circularity. Lemma 3, which states that 'the epimorphic component of γ* is γ* i.e., ((γ*)(a))^o = γ*(a), ∀a∈vC', is asserted without proof and is used in Lemma 4 and in both distributivity chains of Theorem 3 to delete the epimorphic-component operation. This is stronger than Definition 3's requirement that a proper cone have at least one epimorphic component, and it is false in general. Similarly, the statement that 'the ideal categories of Euclidean domains are RR−proper categories' is unsupported and false for Z, since principal ideals of Z are not relatively complemented as a lattice. These are invalid proof steps and false claims, but they are not cases where the theorem's conclusion is built into its inputs by construction. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper rests on newly introduced RR-conditions that are not derived from any prior structure, and on the claim that Euclidean domains satisfy them, which is false. No physical or mathematical constants are fitted.

assumptions (3)

ad hoc to paper RR-conditions: the object set with subobject order is a relatively complemented lattice, and every subset with an upper bound has a unique maximal element.
These conditions are introduced in Definition 5 to make the ring-of-cones construction work; no structural reason is given for why they hold for the intended examples, and the Euclidean domain claim fails.
domain assumption Every morphism in the category has a canonical factorization and every inclusion splits (proper category axioms).
Inherited from Nambooripad's normal categories and the authors' prior work; used throughout Section 3 without proof.
standard math The category is preadditive with a zero object, so each hom-set is an abelian group and composition is bilinear.
Standard background used to define addition of morphisms and the zero cone.

invented entities (2)

RR-proper category
purpose: A category with preadditive structure plus relative-complement lattice and bounded-max axioms, designed so that its proper cones form a ring.
Introduced in Definition 5; no evidence outside the paper that such categories meaningfully arise. The claimed example of Euclidean domains is false.
Ring of proper cones PL(R)
purpose: The purported ring formed by proper cones of the ideal category of a ring.
This construction is the paper's main result; it is not verified independently and its existence for basic rings is in doubt.

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Cite this review

Pith. "Pith review of Ideal categories of rings and ring of cones." pith.science (2026). https://pith.science/paper/VQJJG4XJ

@misc{pith2026190805074,
  author       = {Pith},
  title        = {Pith review of: Ideal categories of rings and ring of cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQJJG4XJ}},
  note         = {Machine review of arXiv:1908.05074}
}

read the original abstract

In this paper we describe the ideal category of a ring R as preadditive proper category. Further it is also shown that the cones in this category is a ring with appropriate addition and multiplication.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Michael Artin, Algebra, Prentice Hall, New Jersey, ISBN 0 − 13 − 004763 − 5(1991)

work page 1991
[2]

A.H.Cliﬀord and G.B.Preston, The algebraic theory of se migroups vol 1 , Math Surveys No.7, American Mathematical Society, Providence, R.I.(1961)

work page 1961
[3]

ISBN 0 − 19 − 851194 − 9(1995)

J.M.Howie, Fundamentals of semigroup theory, Clarendo n Press, Oxford. ISBN 0 − 19 − 851194 − 9(1995)

work page 1995
[4]

S.Mac Lane, Categories for the working mathematician, S pringer Verlag, Newyork, ISBN 0 − 387 − 98403 − 8(1971)

work page 1971
[5]

Mark V Lawson, Inverse semigroups: The theory of partial symmetries, World Scientiﬁc, ISBN 978-981-02-3316-7(1998)

work page 1998
[6]

Sunny Lukose and A.R.Rajan, Ring of normal cones, Indian J. Pure. Appi. Math,41(5):663-681,(2010)

work page 2010
[7]

Barry Mitchell, Rings with several objects, Advances in Mathematics 8 , 1 − 161(1972)

work page 1972
[8]

C.Musili, Introduction to rings and modules, Narosa Pub lishing House, New Delhi, ISBN 81 − 85198 − 64 − 0 (1992)

work page 1992

Show all 12 references

[9]

K.S.S.Nambooripad, Theory of cross connections, Publi cation No.28 - Centre for Mathematical Sciences, Trivandrum(1994)

1994
[10]

P.G.Romeo, Concordant semigroups and balanced catego ries, Southeast Asian Bulletin of Mathematics,31:949-961(2007)

2007
[11]

P.G.Romeo and Sreejamol P.R., Proper Category and its d ual - Proceedings of International Conference on Algebra and Applications, A lligarh Muslim University,UP July, 2018

2018
[12]

New York, ISBN 0 − 12 − 599841 − 4(1988)

Louis Rowen, Ring theory Vol 1, Academic Press, Inc. New York, ISBN 0 − 12 − 599841 − 4(1988) . 1 Assistant Professor, Department of Mathematics, SNM Colleg e, Maliankara, Ernakulam, Kerala, India., 2 Professor, Dept. of Mathe- matics, Cochin University of Science and Technolo...

1988

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Michael Artin, Algebra, Prentice Hall, New Jersey, ISBN 0 − 13 − 004763 − 5(1991)

work page 1991

[2] [2]

A.H.Cliﬀord and G.B.Preston, The algebraic theory of se migroups vol 1 , Math Surveys No.7, American Mathematical Society, Providence, R.I.(1961)

work page 1961

[3] [3]

ISBN 0 − 19 − 851194 − 9(1995)

J.M.Howie, Fundamentals of semigroup theory, Clarendo n Press, Oxford. ISBN 0 − 19 − 851194 − 9(1995)

work page 1995

[4] [4]

S.Mac Lane, Categories for the working mathematician, S pringer Verlag, Newyork, ISBN 0 − 387 − 98403 − 8(1971)

work page 1971

[5] [5]

Mark V Lawson, Inverse semigroups: The theory of partial symmetries, World Scientiﬁc, ISBN 978-981-02-3316-7(1998)

work page 1998

[6] [6]

Sunny Lukose and A.R.Rajan, Ring of normal cones, Indian J. Pure. Appi. Math,41(5):663-681,(2010)

work page 2010

[7] [7]

Barry Mitchell, Rings with several objects, Advances in Mathematics 8 , 1 − 161(1972)

work page 1972

[8] [8]

C.Musili, Introduction to rings and modules, Narosa Pub lishing House, New Delhi, ISBN 81 − 85198 − 64 − 0 (1992)

work page 1992