The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

TL;DR

This paper investigates the structure of the torsion subgroup in the additive group of a specific Lie nilpotent associative ring of class 3, identifying a basis for the elementary abelian 3-group component.

Contribution

It provides an explicit basis for the elementary abelian 3-group component of the additive group of the ring, complementing previous work on the free abelian part.

Findings

Identified a basis for the elementary abelian 3-group B

Described the torsion subgroup structure in detail

Extended understanding of Lie nilpotent associative rings of class 3

Abstract

Let $\mathbb Z \langle X \rangle$ be the free unital associative ring freely generated by an infinite countable set $X = \{ x_1,x_2, \dots \}$. Define a left-normed commutator $[x_1,x_2, \dots, x_n]$ by $[a,b] = ab - ba$, $[a,b,c] = [[a,b],c]$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $\mathbb Z \langle X \rangle$ generated by all commutators $[a_1,a_2, \dots, a_n]$ $( a_i \in \mathbb Z \langle X \rangle )$. Let $T^{(3,2)}$ be the two-sided ideal of the ring $\mathbb Z \langle X \rangle$ generated by all elements $[a_1, a_2, a_3, a_4]$ and $[a_1, a_2] [a_3, a_4, a_5]$ $(a_i \in \mathbb Z \langle X \rangle)$. It has been recently proved in arXiv:1204.2674 that the additive group of $\mathbb Z \langle X \rangle / T^{(4)}$ is a direct sum $ A \oplus B$ where $A$ is a free abelian group isomorphic to the additive group of $\mathbb Z \langle X \rangle / T^{(3,2)}$ and $B = T^{(3,2)} /T^{(4)}$ is an elementary abelian $3$-group. A basis of the free abelian summand $A$ was described explicitly in arXiv:1204.2674. The aim of the present article is to find a basis of the elementary abelian $3$-group $B$.