Quelques résultats sur les anneaux de Lie qui n ' ont pas de chaine infinie de centralisateurs
Samuel Zamour
TL;DR
This paper analyzes Lie rings with no infinite chains of centralizers, introducing the class $\mathfrak{M}_c$ and developing an iterated centralizer framework. It proves the existence and nilpotence of the nilpotent (Fitting) radical $F(L)$ in $\mathfrak{M}_c$ Lie algebras, and establishes an Engel-type theorem in characteristic zero by identifying the Baer radical $B(L)$ with $F(L)$ and employing $\exp(\operatorname{ad}_x)$ to generate inner automorphisms. The results extend group-theoretic model-theoretic methods to Lie rings, including definability aspects in stable settings, where the Fitting radical becomes definable and the nilpotent/solvable radicals exist and are definable in Morley-finite rank scenarios. These findings connect structural properties of iterated centralizers with model-theoretic definability and provide a robust framework for further study of Lie rings under centralizer chain conditions.
Abstract
In line with known results on groups, we show the existence of the nilpotent radical in Lie rings with minimal condition on centralizers. We also prove a form of Engel's theorem if the characteristic is zero.