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:
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)
“The disesteem into which moralists have fallen is due at bottom to their failure to see that in an age like this one the function of the moralist is not to exhort men to be good but to elucidate what the good is. The problem of sanctions is secondary.”
—Walter Lippmann (1889–1974)