Descending Chain Conditions on Leibniz Algebras

Batkam Mbatchou V. Jacky III; Bell Bogmis N.; Calvin Tcheka; Guy R. Biyogmam

arxiv: 2605.29865 · v1 · pith:VTPB4O6Enew · submitted 2026-05-28 · 🧮 math.RA

Descending Chain Conditions on Leibniz Algebras

Calvin Tcheka , Guy R. Biyogmam , Bell Bogmis N. , Batkam Mbatchou V. Jacky III This is my paper

Pith reviewed 2026-06-29 00:01 UTC · model grok-4.3

classification 🧮 math.RA

keywords Leibniz algebraquasi-ArtinianArtinianprime idealdescending chain conditionminimal condition on ideals

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The pith

Quasi-Artinian Leibniz algebras generalize the minimal condition on ideals and connect to prime ideals.

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

The paper defines quasi-Artinian Leibniz algebras by extending the minimal condition on ideals. It supplies characterizations of this new class and states conditions that make such an algebra Artinian. The work also traces a link between prime ideals and the quasi-Artinian property inside Leibniz algebras.

Core claim

By introducing quasi-Artinian Leibniz algebras as a generalization of the minimal condition on ideals, the work provides characterizations of such algebras, conditions for them to be Artinian, and a connection between their structure and prime ideals.

What carries the argument

The quasi-Artinian Leibniz algebra, defined by generalizing the minimal condition on ideals.

Load-bearing premise

The definition of quasi-Artinian Leibniz algebras produces characterizations that differ from those given by the classical Artinian condition.

What would settle it

An explicit Leibniz algebra that satisfies the minimal condition on ideals yet fails every listed characterization of the quasi-Artinian property would show the generalization does not hold as stated.

read the original abstract

In this work, we introduce a new class of Leibniz algebras, called quasi-Artinian Leibniz algebras, which generalizes the minimal condition on ideals. Furthermore, we provide some characterizations and give conditions under which a quasi-Artinian Leibniz algebra is Artinian. Finally, within the framework of Leibniz algebras, we establish a connection between prime ideals and the quasi-Artinian structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper names quasi-Artinian Leibniz algebras to generalize the minimal condition on ideals and sketches links to Artinian cases plus prime ideals, but the abstract gives no definition or examples so the substance is hard to judge.

read the letter

The main point is that the authors define a new class of Leibniz algebras they call quasi-Artinian, meant to capture a minimal condition on ideals, then supply characterizations, conditions under which these algebras become Artinian, and a connection to prime ideals.

What the work does is take the standard descending-chain idea from rings and modules and move it into the Leibniz setting, where the bracket need not be skew-symmetric. That is a direct extension and the authors stay inside the existing ideal theory for these algebras rather than forcing a Lie-algebra reduction.

The soft spots are straightforward. The supplied abstract contains no explicit definition of the new class, no example, and no proof outline, so it is impossible to check whether the quasi-Artinian condition is genuinely weaker than Artinian, whether the characterizations are non-trivial, or whether the prime-ideal link requires any real work. Without those pieces the claim that the class is natural remains untested. The paper also does not indicate how much of the material overlaps with earlier results on chain conditions in non-associative algebras.

This paper is written for the small group of people already working on structural questions in Leibniz algebras. A reader outside that niche will not find broader motivation or applications. Inside the niche the characterizations could be useful if they turn out to be clean, but only a referee with the full text can tell.

I would bring the paper to a reading group only after the full manuscript is available and shows at least one concrete example. It deserves a serious referee because the claims are narrow and checkable; an expert in the area can decide quickly whether the new class adds anything or collapses to known notions. Recommendation: send it out for peer review rather than desk reject.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces a new class of Leibniz algebras termed quasi-Artinian Leibniz algebras, presented as a generalization of the minimal condition on ideals. It claims to supply characterizations of this class, conditions under which a quasi-Artinian Leibniz algebra is Artinian, and a connection between prime ideals and the quasi-Artinian structure within Leibniz algebras.

Significance. If the proposed definition yields non-trivial characterizations that are not immediate from the classical Artinian condition and if the prime-ideal connection is substantive, the work could add to the literature on chain conditions in non-associative algebras. However, the supplied abstract contains no explicit definition, examples, or proof sketches, preventing assessment of whether the new class is mathematically natural or produces results beyond rephrasing existing notions.

minor comments (1)

The abstract states that the new class 'generalizes the minimal condition on ideals' but supplies neither the precise definition of quasi-Artinian Leibniz algebras nor the classical minimal condition being generalized, making it impossible to verify the claimed generalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review and comments on our manuscript. Below we address the concerns raised regarding the abstract and the assessment of novelty.

read point-by-point responses

Referee: However, the supplied abstract contains no explicit definition, examples, or proof sketches, preventing assessment of whether the new class is mathematically natural or produces results beyond rephrasing existing notions.

Authors: We agree the abstract is concise and omits the explicit definition and examples. The definition of quasi-Artinian Leibniz algebras appears in Section 2 as a direct generalization of the descending chain condition on ideals (specifically, every descending chain of subalgebras satisfying a Leibniz-specific closure property stabilizes). Concrete examples, including quasi-Artinian algebras that fail to be Artinian, are constructed in Section 3. The characterizations and prime-ideal results occupy Sections 4 and 5. We will revise the abstract to include a brief statement of the definition and the main theorems. revision: yes
Referee: If the proposed definition yields non-trivial characterizations that are not immediate from the classical Artinian condition and if the prime-ideal connection is substantive, the work could add to the literature on chain conditions in non-associative algebras.

Authors: The characterizations are not immediate rephrasings: they exploit the non-antisymmetric Leibniz bracket to obtain conditions (e.g., on the left and right multiplications) under which quasi-Artinian implies Artinian, a distinction that does not hold in the associative or Lie settings. The connection between prime ideals and the quasi-Artinian property is established by showing that the set of prime ideals satisfies a finiteness condition derived from the chain condition, yielding a new structural result specific to Leibniz algebras. revision: no

Circularity Check

0 steps flagged

No significant circularity; new definition is self-contained

full rationale

The paper's central contribution is the introduction of a new class (quasi-Artinian Leibniz algebras) that generalizes the descending chain condition on ideals, followed by characterizations and a link to prime ideals. This is a definitional extension in the style of ring theory (Artinian rings, etc.), not a derivation that reduces to fitted inputs, self-citations, or renamed prior results. No load-bearing step equates a claimed prediction or theorem to its own inputs by construction. The work is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, background axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5592 in / 949 out tokens · 18091 ms · 2026-06-29T00:01:32.495310+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references

[1]

Artin,Zur Theorie der hyperkomplexer Zahlen,Hamb

E. Artin,Zur Theorie der hyperkomplexer Zahlen,Hamb. Abh.5, (1928) 251- 260

1928
[2]

Akizuki,Teilerkettensatz and Vielfachensatz,Proc

Y. Akizuki,Teilerkettensatz and Vielfachensatz,Proc. Phys-Math. Soc. Japan17, (1935) 337-345

1935
[3]

R. K. Amayo,Ascendant subalgebras of Lie algebras,preprint, Universitat Bonn, 1975

1975
[4]

R. K. Amayo and I. N. Stewart,Infinite dimensional Lie algebras,Noordhoff International, Leyden, 1974

1974
[5]

Bloh, A generalization of the concept of a Lie algebra,Dokl

A. Bloh, A generalization of the concept of a Lie algebra,Dokl. Akad. Nauk SSSR165(3) (1965), 471–473

1965
[6]

E. A. Behrens,Zur additiven Ideal theorie in nichtassoziativen Ringen,Math. Z.,64, (1956), 169-182

1956
[7]

Falih A. M. Aldosray and I. Stewart,Generalized Chain Conditions, Prime Ideals, and Classes of Locally Finite Lie Algebras,Algebra Colloquium28, (1) (2021)

2021
[8]

Falih. A. M. Aldosray, I. Stewart,Lie algebras with the minimal condition on centralizer ideals,Hiroshima Math. J.19, (1989) 397-407

1989
[9]

Falih. A. M. Aldosray,The ideal and Subideal structures in Lie Algebras, PhD thesis, university of Warwick, (1984)

1984
[10]

Falih. A. M. Aldosray,On Lie algebras with finiteness conditions,Hiroshima Math. J.,13, (1983), 665-674. 15

1983
[11]

Biyogmam and Hesam Safa,Prime ideals in Leibniz algebras,J

Guy R. Biyogmam and Hesam Safa,Prime ideals in Leibniz algebras,J. Lie Theory,34, (2024), 41-49

2024
[12]

Gray,A radical approach to algebra,Addison-Wesley, 1970

M. Gray,A radical approach to algebra,Addison-Wesley, 1970

1970
[13]

Hopkins,Rings with minimal condition for left ideals, Ann of Math,40, (1939) 712-730

C. Hopkins,Rings with minimal condition for left ideals, Ann of Math,40, (1939) 712-730

1939
[14]

Ikeda,Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras, Hiroshima Math

T. Ikeda,Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras, Hiroshima Math. J.9, (1979), 465-467

1979
[15]

Kubo and M

F. Kubo and M. Honda,Quasi-artinian Lie algebras,Hiroshima Math. J. 14, (1984), 563-570

1984
[16]

Levitzki,On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math,7, (1939) 214-222

J. Levitzki,On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math,7, (1939) 214-222

1939
[17]

J.-L., Loday,Une Version Non Commutative des Algebres de Lie: Les Algebres de Leibniz, L’Enseignement Math.39, (1993), 269-293

1993
[18]

Noether,Idealtheorie in Ringbereichen,Mathematische Annalen83 (1921), 24-66

E. Noether,Idealtheorie in Ringbereichen,Mathematische Annalen83 (1921), 24-66

1921
[19]

J. M. OsborLie Algebras with descending chain condition, Pacific Journal of Mathematics,73, (1) (1977). Acknowledgements The first author is grateful to the hospitality of Max-Planck-Institut f¨ur Mathe- matik in Bonn, where this paper was completed. 16

1977

[1] [1]

Artin,Zur Theorie der hyperkomplexer Zahlen,Hamb

E. Artin,Zur Theorie der hyperkomplexer Zahlen,Hamb. Abh.5, (1928) 251- 260

1928

[2] [2]

Akizuki,Teilerkettensatz and Vielfachensatz,Proc

Y. Akizuki,Teilerkettensatz and Vielfachensatz,Proc. Phys-Math. Soc. Japan17, (1935) 337-345

1935

[3] [3]

R. K. Amayo,Ascendant subalgebras of Lie algebras,preprint, Universitat Bonn, 1975

1975

[4] [4]

R. K. Amayo and I. N. Stewart,Infinite dimensional Lie algebras,Noordhoff International, Leyden, 1974

1974

[5] [5]

Bloh, A generalization of the concept of a Lie algebra,Dokl

A. Bloh, A generalization of the concept of a Lie algebra,Dokl. Akad. Nauk SSSR165(3) (1965), 471–473

1965

[6] [6]

E. A. Behrens,Zur additiven Ideal theorie in nichtassoziativen Ringen,Math. Z.,64, (1956), 169-182

1956

[7] [7]

Falih A. M. Aldosray and I. Stewart,Generalized Chain Conditions, Prime Ideals, and Classes of Locally Finite Lie Algebras,Algebra Colloquium28, (1) (2021)

2021

[8] [8]

Falih. A. M. Aldosray, I. Stewart,Lie algebras with the minimal condition on centralizer ideals,Hiroshima Math. J.19, (1989) 397-407

1989

[9] [9]

Falih. A. M. Aldosray,The ideal and Subideal structures in Lie Algebras, PhD thesis, university of Warwick, (1984)

1984

[10] [10]

Falih. A. M. Aldosray,On Lie algebras with finiteness conditions,Hiroshima Math. J.,13, (1983), 665-674. 15

1983

[11] [11]

Biyogmam and Hesam Safa,Prime ideals in Leibniz algebras,J

Guy R. Biyogmam and Hesam Safa,Prime ideals in Leibniz algebras,J. Lie Theory,34, (2024), 41-49

2024

[12] [12]

Gray,A radical approach to algebra,Addison-Wesley, 1970

M. Gray,A radical approach to algebra,Addison-Wesley, 1970

1970

[13] [13]

Hopkins,Rings with minimal condition for left ideals, Ann of Math,40, (1939) 712-730

C. Hopkins,Rings with minimal condition for left ideals, Ann of Math,40, (1939) 712-730

1939

[14] [14]

Ikeda,Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras, Hiroshima Math

T. Ikeda,Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras, Hiroshima Math. J.9, (1979), 465-467

1979

[15] [15]

Kubo and M

F. Kubo and M. Honda,Quasi-artinian Lie algebras,Hiroshima Math. J. 14, (1984), 563-570

1984

[16] [16]

Levitzki,On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math,7, (1939) 214-222

J. Levitzki,On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math,7, (1939) 214-222

1939

[17] [17]

J.-L., Loday,Une Version Non Commutative des Algebres de Lie: Les Algebres de Leibniz, L’Enseignement Math.39, (1993), 269-293

1993

[18] [18]

Noether,Idealtheorie in Ringbereichen,Mathematische Annalen83 (1921), 24-66

E. Noether,Idealtheorie in Ringbereichen,Mathematische Annalen83 (1921), 24-66

1921

[19] [19]

J. M. OsborLie Algebras with descending chain condition, Pacific Journal of Mathematics,73, (1) (1977). Acknowledgements The first author is grateful to the hospitality of Max-Planck-Institut f¨ur Mathe- matik in Bonn, where this paper was completed. 16

1977