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The paper proves that every three-dimensional algebra over the real or complex numbers has either infinitely many ideals or at most four, and gives explicit structure-constant conditions that decide which case occurs.

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2026-08-02 20:35 UTC pith:RXGSUZVD

load-bearing objection Some correct elementary tools, but the central dichotomy is unproven: the symmetrization reduction is false and the paper contradicts its own theorem. the 4 major comments →

arxiv 2602.22863 v2 pith:RXGSUZVD submitted 2026-02-26 math.RA

Ideals in Arbitrary Three-Dimensional Algebras

M.V. Velasco , U.A. Rozikov , B.A. Narkuziev This is my paper
classification math.RA MSC 17A1515A2417A30
keywords non-associative algebrasthree-dimensional algebrasidealsstructure constants2-dimensional idealsideal classificationeigenvalue criterioncommutative symmetrization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to settle how many ideals a three-dimensional algebra over the real or complex numbers can have, without assuming associativity. Working from the 27 structure constants of a basis, the authors prove a dichotomy: either the algebra has infinitely many two-dimensional ideals, or it has at most four; in total, either infinitely many ideals or at most four. The proof works by sorting every two-dimensional subspace into one of four basis-dependent types and writing the 'is an ideal' condition for each type as explicit polynomial equations in one or two parameters, so the number of solutions bounds the number of ideals. The authors also characterize one-dimensional ideals as common eigenvectors of six matrices built from the structure constants and give an explicit family attaining two type-IV ideals. If the dichotomy is right, the ideal lattice of any such algebra is extremely constrained—infinite or tiny—which is a sharp structural fact about the lowest dimension where non-associative algebras become genuinely varied.

Core claim

The paper's central discovery is a structure-constant criterion for ideals in a 3-dimensional algebra A with basis e1,e2,e3 and multiplication e_i e_j = sum_k ω_ijk e_k. One-dimensional ideals are exactly lines spanned by common eigenvectors of the six matrices cM_k and fM_k built from the ω_ijk. Two-dimensional ideals are classified into four exclusive types relative to the basis (I: span{e1,e2}; II: span{x e1+e2, e3}; III: span{x e2+e3, e1}; IV: span{x e1+e2, e1+y e3}), and each type's ideal condition becomes explicit polynomial equations in x and y. Solving these systems yields the main dichotomy: either infinitely many 2-dimensional ideals or at most four, and consequently either infinit

What carries the argument

The central object is the cubic structure matrix M=(M1|M2|M3) of 27 constants ω_ijk defined by e_i e_j = Σ_k ω_ijk e_k. From it, the paper forms six linear matrices cM_k and fM_k; a line Ku is a 1-dimensional ideal iff u is a common eigenvector of all six. For two-dimensional ideals, the machinery is the four B-types (I: span{e1,e2}; II: span{x e1+e2, e3}; III: span{x e2+e3, e1}; IV: span{x e1+e2, e1+y e3}); the ideal conditions become systems (4.4) for types II/III and (4.11) for type IV. The number of common solutions (x,y) of (4.11) is controlled by the rank of a 6×7 matrix T|V from (4.17), which is the engine behind the 'infinite or ≤4' dichotomy.

Load-bearing premise

The load-bearing premise is Proposition 3's claim that passing to the symmetrized algebra A^+ is 'not restrictive' for counting two-dimensional ideals; as written the proof gives only that ideals of A are ideals of A^+, while the type-IV analysis assumes every ideal of A^+ is an ideal of A.

What would settle it

Take a non-commutative 3-dimensional algebra over R or C and compute its symmetrized algebra A^+. If some two-dimensional subspace M satisfies A^+ M + M A^+ ⊆ M but fails AM + MA ⊆ M, the reduction in Proposition 3 breaks; a concrete such example with more type-IV ideals than the commutative rank analysis predicts (e.g., three or four) would refute Theorem 6 as stated for arbitrary algebras. Equivalently, search the 27 structure constants for a finite ideal count of 5 or more—the theorem says this cannot happen.

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If this is right

  • For every 3-dimensional algebra over R or C, the ideal lattice has no finite intermediate sizes: if it is not infinite, it has at most four ideals total, and finite counts of one-dimensional ideals are capped at three.
  • The type classification turns ideal-finding into polynomial algebra: one-dimensional ideals are common eigenvectors, and two-dimensional ideals are roots of explicit linear and quadratic systems, so any specific algebra can be checked by computation.
  • Infinite families of ideals are characterized by the normal forms in Theorem 2 and Corollary 2: they occur exactly when the annihilator has dimension at least two or when the multiplication has the listed e3-ei form.
  • The explicit family in Section 7 shows that two ideals of B-type IV can coexist, with rank-4 and rank-5 parameter choices, so the upper bound for that type is sharp at 2.
  • Because the dichotomy is stated for arbitrary non-associative algebras, it would imply that non-associativity does not allow large finite ideal lattices in dimension three; the only escape is infinitely many ideals.

Where Pith is reading between the lines

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

  • The reduction to the symmetrized algebra A^+ is the hinge: the proof of Proposition 3 as printed shows only that ideals of A are ideals of A^+, while the type-IV count needs the converse. Until that converse is supplied, the theorem should be read as established for commutative algebras, with the non-commutative statement conditional on this step.
  • The abstract's claim that the maximum number of 2-dimensional ideals is 2 is stronger than Theorem 6's 'at most four,' and the body does not appear to exhibit 3 or 4 two-dimensional ideals; reconciling these two bounds is either a missing result or a typo.
  • The eigenvector criterion suggests a direct generalization: for an n-dimensional algebra, one-dimensional ideals should correspond to common eigenvectors of 2n matrices built from the structure constants, and 'counting ideals' becomes a problem in common-invariant-subspace computation.
  • A computational search over random structure constants could test the dichotomy directly: for each sampled algebra, solve the polynomial systems for the four B-types and check whether any finite count exceeds four; a counterexample would pinpoint exactly where the symmetrization reduction fails.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies ideals in arbitrary (not necessarily associative) 3-dimensional algebras over R or C. It gives criteria for 1-dimensional ideals (Theorems 1–2, Corollary 2), classifies 2-dimensional subspaces into four types relative to a fixed basis, and attempts to count ideals of each type. The main result (Theorem 6) claims that a 3-dimensional algebra either has infinitely many two-dimensional ideals or at most four, and that if it has finitely many ideals, it has at most four in total. The paper also claims the maximum number of 1-dimensional ideals is 3 and of 2-dimensional ideals is 2, and it presents a class of algebras said to attain the maximum.

Significance. If the main theorem were correct, the paper would contribute a useful classification of ideal lattices in arbitrary 3-dimensional algebras, an area where explicit results are scarce. The four-type decomposition of 2-dimensional subspaces and the use of structure constants are reasonable starting ideas. However, the paper's central reduction to commutative algebras is false, its enumeration of type-II ideals is explicitly incomplete, and the final theorem is asserted without a proof covering the stated generality. The paper does not provide machine-checked proofs or reproducible code, and the internal inconsistencies mean that the main claims are not currently established.

major comments (4)
  1. [§4.1, Proposition 3 (p. 9)] Proposition 3 states that an ideal of the symmetrized algebra A^+ is an ideal of A, but the proof shows only the reverse implication. The asserted implication is false: for the cross-product algebra on R^3 with e1e2=e3, e2e3=e1, e3e1=e2, the symmetrized algebra has zero multiplication and hence infinitely many ideals, while A is the simple Lie algebra so(3) and has no nonzero proper ideals. This invalidates the reduction to commutativity used in §4.4 (Case I.3), §5, and §6. Consequently, Theorem 6 is not proved for arbitrary noncommutative 3-dimensional algebras.
  2. [§4.3, Theorem 5 and Remark 3] Theorem 5 asserts that the number of type-II ideals is exactly 2 iff condition K2 is satisfied and 0 if K1–K3 all fail. Remark 3 immediately concedes that K2 'does not account for all possibilities of (4.4) having two solutions', giving the example of two proportional quadratic equations. Thus the type-II count is incomplete. Since the final bound in Theorem 6 depends on summing the counts of the four types, this incompleteness directly undermines the main theorem.
  3. [Abstract vs. Theorem 6; §7] The abstract states the maximum number of 2-dimensional ideals is 2, while Theorem 6 states the number is at most 4. The introduction claims a family 'achieving the theoretical maximum of four 2-dimensional ideals', but Section 7 constructs algebras with two type-IV ideals and does not exhibit an algebra with four 2-dimensional ideals. Moreover, Theorem 6 is stated without proof: Sections 5 and 6 cover only commutative algebras with a type-I ideal or with two 1-dimensional ideals, and do not address the remaining cases. The main theorem is therefore unsupported.
  4. [§3, Theorem 2 proof] In the proof of Theorem 2, the line 'e_i e_j ≠ 0 for every i,j with i≠j, as I_i ≠ I_j' is wrong. For distinct 1-dimensional ideals I_i and I_j, e_i e_j lies in both I_i and I_j because each is an ideal, so e_i e_j ∈ I_i ∩ I_j = {0}. The subsequent case analysis in (i)⇒(ii), which relies on this assertion and on the product of A being zero, is therefore invalid. This undermines Corollary 2 and the 1-dimensional count that the final theorem would use.
minor comments (4)
  1. [Definition 3(iii)] Item (iii) says 'type II' again; it should read 'type III'.
  2. [Eq. (2.4)] The row vector is printed as (a1 a2 a2); it should be (a1 a2 a3).
  3. [Proposition 3 proof] The proof establishes the converse of the stated claim. If the intended statement is only the inclusion of ideals of A into A^+, that should be stated explicitly, but it would not justify the later reduction.
  4. [Throughout] There are numerous typos and unclear notations, including 'infinitely may' after Definition 2 and inconsistent Greek-letter subscripts in the proof of Theorem 2. A careful editorial pass is needed.

Circularity Check

0 steps flagged

No significant circularity; the ideal-counting derivation is self-contained, with the noted defects being correctness issues rather than circular reasoning.

full rationale

The paper's central claims are derived directly from the definition of an ideal (Definition 1) via explicit structure-constant equations (e.g., (4.4), (4.10), (4.14)) and standard linear algebra. There are no fitted parameters, no quantity is defined in terms of another and then predicted from it, and the paper does not invoke a self-authored uniqueness theorem or a prior ansatz by the same authors. The references cited are standard textbooks and unrelated classification papers; no load-bearing step reduces to a self-citation. The serious mathematical defect in Proposition 3—the symmetrization argument is stated in the wrong direction, so the reduction to commutative algebras in Section 4.4 does not establish Theorem 6 for noncommutative algebras—is a correctness gap, not a circularity: the incorrect implication is not an instance of assuming what one aims to prove, and the counting analysis itself is derived from the definitions rather than from the conclusion. Similarly, the discrepancy between the abstract's claim of at most two 2-dimensional ideals and Theorem 6's bound of four is an internal inconsistency, not a circular step. Accordingly, under the hard rule that circularity requires a demonstrated reduction of the derivation to its own inputs, the honest finding is no significant circularity with score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard linear algebra plus two paper-specific assumptions that the text itself weakens: the commutative reduction (Prop 3, wrong direction) and the K1–K3 completeness (Remark 3 admits K2 incomplete). No fitted parameters or invented entities.

axioms (4)
  • domain assumption The field K is either R or C.
    Stated in Section 2; statements and examples may depend on K (e.g., discriminants >0 vs ≠0 in Condition K2).
  • standard math Capelli–Fontené–Frobenius–Kronecker–Rouché theorem on linear systems.
    Used in Proposition 4 and Corollary 3 for the linearized system (4.14).
  • ad hoc to paper Proposition 3: an ideal of the symmetrized algebra A^+ of a given B-type is an ideal of A.
    Used to reduce to commutative algebras; false as stated, and the proof shows only the reverse inclusion.
  • ad hoc to paper Completeness of conditions K1–K3 for counting type-II ideals.
    Theorem 5's N_II(A) classification depends on it; Remark 3 explicitly admits K2 is incomplete.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Ideals in Arbitrary Three-Dimensional Algebras." pith.science (2026). https://pith.science/paper/RXGSUZVD

@misc{pith2026260222863,
  author       = {Pith},
  title        = {Pith review of: Ideals in Arbitrary Three-Dimensional Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXGSUZVD}},
  note         = {Machine review of arXiv:2602.22863}
}
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read the original abstract

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all ideals of A, providing an explicit characterization of both 1-dimensional and 2-dimensional ideals. Moreover, we classify 2-dimensional ideals into 4 distinct types. We prove that A either has infinitely many ideals or at most 4. We also show that, in any case, the maximum number of 1-dimensional ideals is 3, while the maximum number of 2-dimensional ideals is 2. Finally, we present a class of algebras with a finite number of ideals that attain this theoretical maximum.

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Works this paper leans on

53 extracted references · 1 linked inside Pith

  1. [1]

    A. A. Albert, Non-associative algebras, Ann. Math. 43, 1942, 685-707

    1942

  2. [2]

    K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative, Academic Press, 1982

    1982

  3. [3]

    Jacobson, Structure and representations of Jordan algebras, AMS, 1968

    N. Jacobson, Structure and representations of Jordan algebras, AMS, 1968

    1968

  4. [4]

    Zorn, Finite-dimensional non-associative algebras and Lie algebras, Bull

    P. Zorn, Finite-dimensional non-associative algebras and Lie algebras, Bull. AMS 42 (1936), 569-574

    1936

  5. [5]

    I. N. Herstein, Noncommutative rings , American Mathematical Soc., 1994

    1994

  6. [6]

    Serre, Lie algebras and Lie groups, Springer, 1992

    J.-P. Serre, Lie algebras and Lie groups, Springer, 1992

    1992

  7. [7]

    S. M. Sidorov et al., *Classification of 3-dimensional algebras*, J. Algebra **281** (2004), 251–262

    2004

  8. [8]

    Kobayashi, K

    Y. Kobayashi, K. Shirayanagi, M. Tsukada, and S. Takahasi, A complete classification of three-dimensional algebras over R and C . Asian-European Journal of Mathematics, 14(08), 2150131, (2021)

    2021

  9. [9]

    A. A. Slin’ko, *On 3-dimensional Malcev algebras*, Comm. Algebra **30** (2002), 1477–1487

    2002

  10. [10]

    I. R. Shafarevich, Basic notions of algebra, Springer, 2005

    2005

  11. [11]

    L. A. Bokut et al., *Gröbner-Shirshov bases for 3-dimensional Lie algebras*, J. Symb. Comput. **45** (2010), 1253–1272

    2010

  12. [12]

    M. R. Bremner et al., *Ideals in non-associative algebras: A survey*, Contemp. Math. **672** (2016), 1–18

    2016

  13. [13]

    Kaygorodov, M

    I. Kaygorodov, M. Khrypchenko, P. P\'aez-Guill\'an, The geometric classification of non-associative algebras: a survey. Communications in Mathematics, 32(2), 2024, 185-284

    2024

  14. [14]

    I. P. Shestakov, *Non-associative algebras: New trends and applications*, J. Math. Sci. **161** (2009), 769–780

    2009

  15. [15]

    Nijenhuis, R

    A. Nijenhuis, R. W. Richardson, Deformations of Lie algebra structures, Journal of Mathematics and Mechanics, 17, 1967, 89-105

    1967

  16. [16]

    Liebmann, H

    M. Liebmann, H. Ruhaak, B. Henschenmacher, Non-Associative Algebras and Quantum Physics -- A Historical Perspective, arXiv:1909.04027. 2019

    Pith/arXiv arXiv 1909

  17. [17]

    Abdalla, Mihir Bellare, et

    M. Abdalla, Mihir Bellare, et. al, \ Searchable

  18. [18]

    R. Bai, H. Liu, M. Zhang, 3-Lie algebras realized by

  19. [19]

    Casas, M

    J.M. Casas, M. Ladra, U.A. Rozikov, A chain of

  20. [20]

    Casas, M

    J.M. Casas, M. Ladra, U.A. Rozikov, Markov processes

  21. [21]

    Craig, Understanding Augmented Reality: Concepts

    A.B. Craig, Understanding Augmented Reality: Concepts

  22. [22]

    Davenport, \ A vector approach to the algebra of

    P.B. Davenport, \ A vector approach to the algebra of

  23. [23]

    Deshpande, M.S

    A.M. Deshpande, M.S. Deshpande and D.N. Kayatanavar,

  24. [24]

    Edward Angel and Dave Shreiner, \ Interactive Computer

  25. [25]

    Francesco Regazzoni, Bodhisatwa Mazumdar, Sri

  26. [26]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, A. Courville, Deep

  27. [27]

    Hartley, A

    R. Hartley, A. Zisserman, Multiple View Geometry in

  28. [28]

    Hughes, A

    J.F. Hughes, A. Van Dam, M. McGuire, D.F. Sklar, J.D

  29. [29]

    A. L. Gorodentsev, Algebra I: Textbook for Students of Mathematics, Springer, 2016

    2016

  30. [30]

    Imomkulov, M.V

    A.N. Imomkulov, M.V. Velasco, Chain of

  31. [31]

    Jaydip Sen, Cryptography and Security in Computing,

  32. [32]

    Jacobson, Lie Algebras, Interscience Tracts in

    N. Jacobson, Lie Algebras, Interscience Tracts in

  33. [33]

    Kiran, et

    S. Kiran, et. al., Analysis of in-place matrix rotation

  34. [34]

    Ladra, U.A

    M. Ladra, U.A. Rozikov, Flow of finite-dimensional

  35. [35]

    Ladra, U.A

    M. Ladra, U.A. Rozikov, Construction of flows of

  36. [36]

    Ladra, U.A

    M. Ladra, U.A. Rozikov, Algebras of cubic matrices,

  37. [37]

    Maksimov, Cubic stochastic matrices and their

    V.M. Maksimov, Cubic stochastic matrices and their

  38. [38]

    J. C. Marcos, M. Victoria Velasco, The multiplicative

  39. [39]

    Narkuziyev, U.A

    B.A. Narkuziyev, U.A. Rozikov, Classification in chains

  40. [40]

    Shah, Foram A

    Nita H. Shah, Foram A. Thakkar , \ Matrix and

  41. [41]

    Omirov, U.A

    B.A. Omirov, U.A. Rozikov, K.M. Tulenbayev, On

  42. [42]

    Qin Liu, Guojun Wang, Jie Wu, Time-based proxy

  43. [43]

    Riley, M.P

    K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical

  44. [44]

    Rozikov, Population dynamics: algebraic and

    U.A. Rozikov, Population dynamics: algebraic and

  45. [45]

    Rozikov, Sh.N

    U.A. Rozikov, Sh.N. Murodov, Dynamics of two-dimensional

  46. [46]

    Rozikov, M.V

    U.A. Rozikov, M.V. Velasco, B.A. Narkuziev, \

  47. [47]

    Su, et al., Roformer: enhanced transformer with

    J. Su, et al., Roformer: enhanced transformer with

  48. [48]

    Suetens, Fundamentals of Medical Imaging

    P. Suetens, Fundamentals of Medical Imaging

  49. [49]

    Tarantola, Inverse Problem Theory and Methods

    A. Tarantola, Inverse Problem Theory and Methods

  50. [50]

    Tuecke et al., Globus auth: A research identity

    S. Tuecke et al., Globus auth: A research identity

  51. [51]

    Victoria Velasco, Spectral theory for non-associative

    M. Victoria Velasco, Spectral theory for non-associative

  52. [52]

    Y. Wang, T. Jiang, S. Ma, and W. Gao, \ Novel

  53. [53]

    Y. Zhou, C. Barnes, J. Lu, J. Yang and H. Li, On the

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.