Lie rings in finite-dimensional theories

Abstract

We study Lie rings definable in a finite-dimensional theory, extending the results for the finite Morley rank case. In particular, we prove a classification of Lie rings of dimension up to four in the NIP or connected case. In characteristic $0$, we verify a version of the Cherlin-Zilber Conjecture. Moreover, we characterize the actions of some classes, namely abelian, nilpotent and soluble, of Lie rings of finite dimension. Finally, we show the existence of definable envelopes for nilpotent and soluble Lie rings. These results are used to verify that the Fitting and the Radical ideal of a Lie ring of finite dimension are both definable and respectively nilpotent and soluble.