Group Ring - Group Rings Over An Infinite Group

Group Rings Over An Infinite Group

Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research. The case where R is the field of complex numbers is probably the one best studied. In this case, Irving Kaplansky proved that if a and b are elements of C with ab = 1, then ba = 1. Whether this is true if R is a field of positive characteristic remains unknown.

A long-standing conjecture of Kaplansky (~1940) says that if G is a torsion-free group, and K is a field, then the group ring K has no non-trivial zero divisors. This conjecture is equivalent to K having no non-trivial nilpotents under the same hypotheses for K and G.

In fact, the condition that K is a field can be relaxed to any ring that can be embedded into an integral domain.

The conjecture remains open in full generality, however some special cases of torsion-free groups have been shown to satisfy the zero divisor conjecture. These include:

• Unique product groups (which include virtually abelian groups, orderable groups, and free groups, since they are orderable)

• Elementary amenable groups

• Diffuse groups - in particular, groups that act freely isometrically on R-trees, and the fundamental groups of surface groups except for the fundamental groups of direct sums of one, two or three copies of the projective plane.

The case of G being a topological group is discussed in greater detail in the article on group algebras.