Some Varieties of Lie Rings

TL;DR

This paper generalizes Macdonald's theorems from nilpotent groups to Lie rings, establishing bounds on nilpotency and solvability for specific varieties of Lie rings across different characteristics.

Contribution

It introduces new theorems for varieties of Lie rings, extending known results from group theory to the Lie ring context, with explicit bounds on nilpotency and solvability.

Findings

Lie rings in variety (1.1) are nilpotent of exponent ≤ n+2

Lie rings in variety (1.2) have their squares nilpotent of exponent ≤ n+1

Lie rings in variety (1.3) are solvable of length ≤ n+1

Abstract

In this paper, several theorems of Macdonald \cite{Mac1961,Mac1962} on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some equations (see Eqs. (\ref{eq:1.1})-(\ref{eq:1.3}) below). We prove that each Lie ring in variety $(\ref{eq:1.1})$ is nilpotent of exponent at most $n+2$; if $L$ is a Lie ring in variety $(\ref{eq:1.2})$, then $L^{2}$ is nilpotent of exponent at most $n+1$; and each Lie ring in variety $(\ref{eq:1.3})$ is solvable of length at most $n+1$. Finally, we also discuss some varieties of solvable Lie rings and the varieties of Lie rings defined by the properties of subrings.