Torsion-free soluble groups, completions, and the zero divisor conjecture

Crawley-Boevey, W.W., Kropholler, P.H. and Linnell, P.A. (1988) Torsion-free soluble groups, completions, and the zero divisor conjecture. Journal of Pure and Applied Algebra, 54 (2-3), 181-196. (doi:10.1016/0022-4049(88)90029-1).

Abstract

This paper contains two results which bear upon the zero-divisor conjecture for group rings. The first, proved using commutative algebra, asserts that a finitely generated torsion-free meta- belian-by-finite group has many torsion-free quotients of finite rank. The second result concerns the completion of the group algebra kG at its augmentation ideal when G is a polycyclic pro-p group and k is an algebraically closed field of characteristics p>0. For example, if G is torsion-free it is shown that this completion is a domain. These two results imply that if G is a torsion-free soluble group of derived length at most three, and K is a field of characteristics zero, then KG is a domain.

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Contributors

Author: P.H. Kropholler

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