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

Abstract

Let $\mathbb{Z}⟨X⟩$ 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 $\mathbb{Z}⟨X⟩$ generated by all commutators $[a_1,a_2,\dots ,a_n]$ $(a_i\in \mathbb{Z}⟨X⟩)$. Let $T^{(3,2)}$ be the two-sided ideal of the ring $\mathbb{Z}⟨X⟩$ generated by all elements $[a_1,a_2,a_3,a_4]$ and $[a_1,a_2][a_3,a_4,a_5]$ $(a_i\in \mathbb{Z}⟨X⟩)$. It has been recently proved in arXiv:1204.2674 that the additive group of $\mathbb{Z}⟨X⟩/T^{(4)}$ is a direct sum $A\oplus B$ where $A$ is a free abelian group isomorphic to the additive group of $\mathbb{Z}⟨X⟩/T^{(3,2)}$ 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 $B$.

23 pages; extended introduction, additional references

References in corpus (5)

• The additive group of a Lie nilpotent associative ring
• Lower central series of a free associative algebra over the integers and finite fields
• Relations in universal Lie nilpotent associative algebras of class 4
• On $[A,A]/[A,[A,A]]$ and on a $W_n$-action on the consecutive commutators of free associative algebra
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Cited by in corpus (6)

• Products of commutators in a Lie nilpotent associative algebra
• Relations in universal Lie nilpotent associative algebras of class 4
• On Lower Central Series Quotients of Finitely Generated Algebras over $\mathbb{Z}$
• On some products of commutators in an associative ring
• Products of several commutators in a Lie nilpotent associative algebra
• On Lie nilpotent associative algebras