The additive group of a Lie nilpotent associative ring

Details The additive group of a Lie nilpotent associative ring The additive group of a Lie nilpotent associative ring

2012-04-12

arxiv.org/abs/1204.2674v1

Mathematics

Rings and Algebras

Scientific paper

Let Z be the free unitary associative ring on the 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 generated by all commutators [a_1,a_2,...,a_n] (a_i \in Z). It can be easily seen that the additive group of the quotient ring Z/T^(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have proved that the additive group of Z/T^(3) is free abelian as well. In the present note we show that this is not the case for Z/T^(4). More precisely, let v = [x_1,x_2,x_3][x_4,x_5]; we prove that 3v \in T^(4) but v \notin T^(4). Thus, the additive group of the quotient ring Z/T^(4) contains elements of order 3.

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