Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4585099 | Journal of Algebra | 2013 | 13 Pages |
Abstract
Let ZãXã be the free unital associative ring freely generated by an infinite countable set X={x1,x2,â¦}. Define a left-normed commutator [x1,x2,â¦,xn] by [a,b]=abâba, [a,b,c]=[[a,b],c]. For n⩾2, let T(n) be the two-sided ideal in ZãXã generated by all commutators [a1,a2,â¦,an] (aiâZãXã). It can be easily seen that the additive group of the quotient ring ZãXã/T(2) is a free abelian group. Recently Bhupatiraju, Etingof, Jordan, Kuszmaul and Li have noted that the additive group of ZãXã/T(3) is also free abelian. In the present note we show that this is not the case for ZãXã/T(4). More precisely, let T(3,2) be the ideal in ZãXã generated by T(4) together with all elements [a1,a2,a3][a4,a5] (aiâZãXã). We prove that T(3,2)/T(4) is a non-trivial elementary abelian 3-group and the additive group of ZãXã/T(3,2) is free abelian.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Alexei Krasilnikov,