The characteristic-3 case of the classification of Lie rings of finite Morley rank: Rosengarten's Fact 1 as an obstruction, and simple Lie rings of Morley rank 4 in characteristic 3

Deloro and Tindzogho Ntsiri proved the Lie-ring analogue of the Cherlin–Zilber conjecture in characteristic 0 (every simple Lie ring of finite Morley rank is a Lie algebra of Lie-Chevalley type over a definable field, in any rank) and showed that in characteristic other than 2 and 3 a simple Lie ring cannot have Morley rank 4. The entire edifice rests on Rosengarten's Fact 1: every infinite Lie ring of finite Morley rank and characteristic other than 2, 3 contains an infinite, definable, connected, abelian subring — a result whose proof relies on finite-case Lie-algebra inputs and is open in characteristic 3. The problem, stated as the first question of the source paper ('What happens to our theorem in characteristic 3?'), is to settle the characteristic-3 case of that theorem: since the characteristic-0 clause concerns only characteristic 0, what is at issue in characteristic 3 is the rank-4 clause, namely whether a simple Lie ring of finite Morley rank and characteristic 3 can have Morley rank 4. The source records the key obstruction in the same breath: Fact 1 is open in characteristic 3. The characteristic-3 case of the question remains open.

Background

A Lie ring is an abelian group $\mathfrak{g}=(\mathfrak{g};+)$ equipped with a bi-additive, antisymmetric bracket $[\cdot,\cdot]$ satisfying the Jacobi identity; subrings, ideals, solubility, nilpotence, and simplicity are defined purely in these terms, so a Lie ring carries no a priori field or linear structure. The characteristic of a Lie ring is the exponent of its underlying additive group ('characteristic 0' means torsion-free divisible); every simple Lie ring has characteristic 0 or a prime, and is therefore a simple Lie algebra over $\mathbb{Q}$ or $\mathbb{F}_p$, but of possibly infinite dimension. The phrase Lie algebra is reserved, following the source paper, for the case where an infinite definable field acts compatibly.

Morley rank is the model-theoretic notion of dimension: a subset of $\mathfrak{g}^n$ is definable if it is the solution set of a first-order formula in the language of the additive group and bracket (with parameters), and $\mathfrak{g}$ has finite Morley rank if there is an integer-valued dimension on definable sets obeying the usual algebraic rules (dimension of a union is the maximum; dimension is additive over definable maps with uniform fibre dimension; 'finite' coincides with 'dimension 0'). This is the standard tameness hypothesis of model-theoretic algebra: it abstracts the dimension theory of algebraic geometry while assuming no geometry. Definable subgroups have a smallest definable subgroup of finite index, the connected component $H^\circ$; 'connected' means $H=H^\circ$. A definable field is a field whose domain and operations are definable in the ambient structure.

The group-theoretic Cherlin–Zilber conjecture predicts that every infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field; it remains open. The source paper studies the Lie-ring counterpart, the 'log CZ' conjecture (implicit in Rosengarten's 1991 thesis): every infinite simple Lie ring of finite Morley rank of sufficiently large characteristic is a simple Lie algebra over an algebraically closed field. The algebraic target is understood: by the Block–Premet–Strade–Wilson classification (see the survey of Premet and Strade), a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic at least 5 is either of Chevalley type (obtained from a simple root system $\Phi$ and a Chevalley basis, i.e. a scalar extension $\mathfrak{G}_\Phi(R)$ of the integral Chevalley ring — 'Lie-Chevalley type' in the source's terminology) or of Cartan type (Witt, special, Hamiltonian, contact algebras, which become finite-dimensional exactly in positive characteristic), with an additional Melikian family in characteristic 5. Modular Lie algebra theory is, however, far less complete below characteristic 5.

What is known for Lie rings of finite Morley rank, always in characteristic $\neq 2,3$ unless stated: Rosengarten proved that a connected Lie ring of Morley rank 1 is abelian; a connected Lie ring of Morley rank 2 is soluble, and if non-nilpotent then $\mathfrak{g}/Z(\mathfrak{g})\simeq\mathfrak{ga}_1(\mathbb{K})$, the 2-dimensional non-abelian Lie algebra with basis $x,y$ and $[x,y]=y$, over an algebraically closed definable field; and a connected non-soluble Lie ring of Morley rank 3 satisfies $\mathfrak{g}/Z(\mathfrak{g})\simeq\mathfrak{sl}_2(\mathbb{K})$, the Lie algebra of $2\times 2$ matrices of trace zero. His foundational tool, Fact 1 of the source paper, states that every infinite Lie ring of finite Morley rank and characteristic $\neq 2,3$ contains an infinite, definable, connected, abelian subring (the analogue of Reineke's theorem on definably minimal groups); the known proof is non-trivial and reduces to results on finite Lie algebras. Deloro and Tindzogho Ntsiri then proved the main theorem: a simple Lie ring of finite Morley rank of characteristic 0 is a Lie algebra of Lie-Chevalley type over a definable field (any rank), and in characteristic $\neq 2,3$ there are no simple Lie rings of Morley rank 4.

The method matters for what breaks in characteristic 3. Since a simple Lie ring of finite Morley rank need not be finite-dimensional over anything, the proof cannot use Cartan subalgebras or gradings; it proceeds by local analysis of Borel subrings (definable, connected, soluble subrings, maximal as such), whose existence and quasi-self-normalisation (each has finite index in its normaliser) are derived from Fact 1, together with a 'linearisation in dimension 1' lemma which also depends on Fact 1. Every clause of the source's rank-4 analysis is built on this foundation, and Fact 1's proof relies on finite-case Lie-algebra results that are unavailable in characteristic 3, where moreover the modular classification itself requires characteristic at least 5. Characteristic 2 is a genuinely different regime — known model-theoretic constructions recorded in the source paper (Baudisch's construction) produce non-algebraic nilpotent definable Lie rings of Morley rank 2 there — and is deliberately outside the source question, which concerns characteristic 3 alone.

Problem Statement

Resolve Question 1 of Deloro–Tindzogho Ntsiri (2024), posed in the source's §4 ('Appendix: questions') under the heading 'Extending the result': 'What happens to our theorem in characteristic 3?' The setting is that of the source paper — Lie rings equipped only with the first-order definable structure coming from the additive group and the Lie bracket, measured by Morley rank. 'Our theorem' is the source's main theorem: a simple Lie ring $\mathfrak{g}$ of finite Morley rank is, in characteristic 0, a Lie algebra of Lie-Chevalley type over a definable field (any rank), and, in characteristic $\neq 2,3$, does not have Morley rank 4. The characteristic-0 clause says nothing in characteristic 3, so the question concerns the rank-4 clause:

Determine whether a simple Lie ring of finite Morley rank and characteristic 3 can have Morley rank 4 — that is, either prove that no such Lie ring exists, extending the source's theorem to characteristic 3, or exhibit one.

The source records the key obstruction in the same question: Rosengarten's Fact 1 (every infinite Lie ring of finite Morley rank and characteristic $\neq 2,3$ contains an infinite, definable, connected, abelian subring) is open in characteristic 3, and in the source's proof Borel-subring existence, quasi-self-normalisation, and dimension-1 linearisation are all derived from it. Settling Fact 1 in characteristic 3 would therefore be substantial partial progress on Question 1, but it is the obstruction the source identifies rather than a further deliverable added to it: a proof that no simple Lie ring of Morley rank 4 and characteristic 3 exists, or an explicit example of such a Lie ring, answers Question 1 in either direction.

Scope notes, following the source. Rosengarten's rank 1–3 results recalled in the Background (rank 1 abelian; rank 2 soluble, and if non-nilpotent covering $\mathfrak{ga}_1(\mathbb{K})$; rank 3 non-soluble covering $\mathfrak{sl}_2(\mathbb{K})$) are stated in the source only for characteristic $\neq 2,3$, and no question of the source's §4 asks whether they extend to characteristic 3; they are recorded here as background, not as targets of this problem. Characteristic 2 is excluded from the question, as in the source: it is the prime where definable pathologies of Morley rank 2 are known to exist, and the source asks only about characteristic 3.

References

1. Adrien Deloro and Jules Tindzogho Ntsiri, 'Simple Lie rings of Morley rank 4 (The Spanish Inquisition)', Journal of Algebra 651 (2024) 243–280, DOI 10.1016/j.jalgebra.2024.04.012; preprint arXiv:2309.12465, https://arxiv.org/abs/2309.12465
2. Adrien Deloro and Jules Tindzogho Ntsiri, 'Soluble Lie rings of finite Morley rank', Model Theory 4 (2025), no. 2, 163–175, DOI 10.2140/mt.2025.4.163; preprint arXiv:2409.07783, https://arxiv.org/abs/2409.07783
3. Jules Tindzogho Ntsiri and Samuel Zamour, 'Cartan subrings in soluble ranked Lie rings', preprint arXiv:2511.20323 [math.LO], v1 November 2025, v2 April 2026, https://arxiv.org/abs/2511.20323
4. Moreno Invitti, 'Lie rings in finite-dimensional theories', preprint arXiv:2511.06068 [math.LO], November 2025, https://arxiv.org/abs/2511.06068
5. Samuel Zamour, 'Quelques résultats sur les anneaux de Lie qui n'ont pas de chaine infinie de centralisateurs', preprint arXiv:2503.06230 [math.LO], 2025, https://arxiv.org/abs/2503.06230
6. Samuel Zamour, 'On p-Lie algebras of finite Morley rank', preprint arXiv:2605.13700 [math.LO], v1 May 2026, v2 July 2026, https://arxiv.org/abs/2605.13700
7. Alexandre Borovik, 'Finite group actions on abelian groups of finite Morley rank', Model Theory 3 (2024) 539–569, DOI 10.2140/mt.2024.3.539, https://doi.org/10.2140/mt.2024.3.539
8. Alexander Premet and Helmut Strade, 'Classification of finite dimensional simple Lie algebras in prime characteristics', preprint arXiv:math/0601380, 2006, https://arxiv.org/abs/math/0601380
9. Richard Rosengarten, 'Aleph-zero stable Lie algebras', PhD thesis, Rutgers, The State University of New Jersey (New Brunswick), 1991, 55 pp.; unpublished thesis, no DOI, ISBN, or arXiv entry exists