The additive group of a Lie nilpotent associative ring

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.