Simple Lie rings of Morley rank 4

TL;DR

This paper proves the Lie ring analogue of the Cherlin-Zilber conjecture for certain ranks and characteristics, advancing understanding of the structure of simple Lie rings of Morley rank 4.

Contribution

It establishes the conjecture for Lie rings of Morley rank 4 in characteristic not 2 or 3, filling a gap in the classification of simple Lie rings.

Findings

Proves the conjecture for characteristic 0 and all ranks.

Proves the conjecture for characteristic not 2, 3, and rank ≤ 4.

Advances the classification of simple Lie rings of Morley rank 4.

Abstract

We prove the Lie ring equivalent of the Cherlin-Zilber conjecture -- in characteristic 0, for any rank and -- in characteristic not 2, 3 for rank less than or equal to 4. Both are open in the group case.