The additive group of a Lie nilpotent associative ring

TL;DR

This paper investigates the structure of additive groups of certain quotient rings derived from free associative rings with Lie nilpotency conditions, revealing new algebraic properties and differences at higher levels of commutator ideals.

Contribution

It demonstrates that the additive group of Z<X> /T^(4) is not free abelian, unlike lower cases, and characterizes the structure of related quotient groups, showing a non-trivial elementary abelian 3-group.

Findings

The additive group of Z<X> /T^(2) is free abelian.

The additive group of Z<X> /T^(3) is free abelian.

The additive group of Z<X> /T^(4) is not free abelian, containing a non-trivial elementary abelian 3-group.

Abstract

Let Z<X> be the free unitary associative ring freely generated by an infinite countable set X = {x_1, x_2,...}. Define a left-normed commutator [x_1, x_2, ..., x_n] by [a,b] = ab - ba, [a,b,c] = [[a,b],c]. For n \ge 2, let T^(n) be the ideal in Z<X> generated by all commutators [a_1,a_2,..., a_n] (a_i \in Z<X>). It can be easily seen that the additive group of the quotient ring Z<X> /T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of Z<X> /T^(3) is free abelian as well. In the present note we show that this is not the case for Z<X> /T^(4). More precisely, let T^(3,2) be the ideal in Z<X> generated by T^(4) together with all elements [a_1, a_2, a_3][a_4, a_5] (a_i \in Z<X>). We prove that T^(3,2)/T^(4) is a non-trivial elementary abelian 3-group and the additive group of Z<X> /T^(3,2) is free abelian.