Editing Group ring (section)

==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.<ref>{{cite journal|author=Passman, Donald S.|author-link=Donald S. Passman|title=What is a group ring?|journal=Amer. Math. Monthly|volume=83|year=1976|issue=3 |pages=173–185|url=http://www.maa.org/programs/maa-awards/writing-awards/what-is-a-group-ring|doi=10.2307/2977018|jstor=2977018 }}</ref> 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'''[''G''] with {{nowrap|1=''ab'' = 1}}, then {{nowrap|1=''ba'' = 1}}. Whether this is true if ''R'' is a field of positive characteristic remains unknown.

A long-standing [[Kaplansky's conjectures|conjecture of Kaplansky]] (~1940) says that if ''G'' is a [[torsion-free group]], and ''K'' is a field, then the group ring ''K''[''G''] has no non-trivial [[zero divisor]]s. This conjecture is equivalent to ''K''[''G''] having no non-trivial [[nilpotent]]s 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 (e.g. [[orderable group]]s, in particular [[free group]]s)
* [[Elementary amenable group]]s (e.g. [[virtually abelian group]]s)
* 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 where ''G'' is a [[topological group]] is discussed in greater detail in the article [[Group algebra of a locally compact group]].