Vol. 90, No. 1, 1980

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The zero divisor conjecture for some solvable groups

Robert L. Snider

Vol. 90 (1980), No. 1, 191–196

Let F be a field and G a group. The zero divisor conjecture states that if G is torsion free, then the group algebra F[G] is torsion free. A series of papers by various authors have resulted in a proof of this conjecture for polycyclic-by-finite groups. The next most natural step would seem to be groups which are poly-(torsion free rank one abelian)-by-finite. These are precisely the solvable groups of finite cohomological dimension. A perhaps more attractive description of these groups is the solvable-by-finite subgroups of GLn(Q), Q being the rational numbers. We are able to prove this conjecture for the class of these groups where the primes in the finite “top” are different from the primes that make the rank one abelian factors non finitely generated. The key ingredient in the proof is a localization theorem which makes these non-Noetherian group rings Noetherian.

Mathematical Subject Classification 2000
Primary: 16A27, 16A27
Secondary: 20C07
Received: 20 February 1979
Published: 1 September 1980
Robert L. Snider