Group Rings and The Zero Divisor Problem
Suppose that G is a group and K is a field. Is the group ring R = K a domain? The identity
shows that an element g of finite order n induces a zero divisor 1−g in R. The zero divisor problem asks whether this is the only obstruction, in other words,
- Given a field K and a torsion-free group G, is it true that K contains no zero divisors?
No countexamples are known, but the problem remains open in general (as of 2007).
For many special classes of groups, the answer is affirmative. Farkas and Snider proved in 1976 that if G is a torsion-free polycyclic-by-finite group and char K = 0 then the group ring K 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 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 pth congruence subgroup of GL(n,Z).
Read more about this topic: Domain (ring Theory)
Famous quotes containing the words group, rings and/or problem:
“It is not God that is worshipped but the group or authority that claims to speak in His name. Sin becomes disobedience to authority not violation of integrity.”
—Sarvepalli, Sir Radhakrishnan (1888–1975)
“If a man do not erect in this age his own tomb ere he dies, he shall live no longer in monument than the bell rings and the widow weeps.”
—William Shakespeare (1564–1616)
“The government is huge, stupid, greedy and makes nosy, officious and dangerous intrusions into the smallest corners of life—this much we can stand. But the real problem is that government is boring. We could cure or mitigate the other ills Washington visits on us if we could only bring ourselves to pay attention to Washington itself. But we cannot.”
—P.J. (Patrick Jake)