# Torsion in free centre-by-nilpotent-by-abelian Lie rings of rank 2

## TL;DR

This paper characterizes the torsion subgroup in the additive group of a specific class of free Lie rings constructed from two generators, focusing on cases where the nilpotency class is a prime number.

## Contribution

It provides a complete description of torsion in free centre-by-nilpotent-by-abelian Lie rings of rank 2 for prime nilpotency classes.

## Key findings

- Identifies the torsion subgroup structure for the case |X|=2 and c prime.
- Provides explicit descriptions of torsion elements.
- Advances understanding of torsion phenomena in specific Lie ring constructions.

## Abstract

For $c\geq 2$, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set $X$ is the quotient $L/[(L')^c,L]$ where $L$ is the free Lie ring on $X$, and $(L')^c$ denotes the $c$th term of the lower central series of the derived ideal $L'=L^2$ of $L$. In this paper we give a complete description of the torsion subgroup of its additive group in the case where $|X|=2$ and $c$ is a prime number.