Editing Domain (ring theory) (section)

== Group rings and the zero divisor problem ==

Suppose that ''G'' is a [[group (mathematics)|group]] and ''K'' is a [[field (mathematics)|field]]. Is the [[group ring]] {{nowrap|1=''R'' = ''K''[''G'']}} a domain?  The identity

: <math> (1-g)(1+g+\cdots+g^{n-1})=1-g^n,</math>

shows that an element ''g'' of finite [[order (group theory)|order]] {{nowrap|''n'' > 1}} induces a zero divisor {{nowrap|1 − ''g''}} in ''R''. The '''zero divisor problem''' asks whether this is the only obstruction; in other words,

: Given a [[field (mathematics)|field]] ''K'' and a [[torsion-free group]] ''G'', is it true that ''K''[''G''] contains no zero divisors?

No counterexamples are known, but the problem remains open in general (as of 2017).

For many special classes of groups, the answer is affirmative. Farkas and Snider proved in 1976 that if ''G'' is a torsion-free [[polycyclic group|polycyclic-by-finite]] group and {{nowrap|1=char ''K'' = 0}} then the group ring ''K''[''G''] is a domain. Later (1980) Cliff removed the restriction on the characteristic of the field. In 1988, Kropholler, Linnell and Moody generalized these results to the case of torsion-free [[solvable group|solvable]] and solvable-by-finite groups. Earlier (1965) work of [[Michel Lazard]], whose importance was not appreciated by the specialists in the field for about 20 years, had dealt with the case where ''K'' is the ring of [[p-adic integers]] and ''G'' is the ''p''th [[congruence subgroup]] of {{nowrap|GL(''n'', '''Z''')}}.