The identity shows that an element g of finite order n > 1 induces a zero divisor 1 − g in R. The zero divisor problem asks whether this is the only obstruction; in other words, No counterexamples are known, but the problem remains open in general (as of 2017).
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).
Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain if and only if it is reduced and its spectrum Spec R is an irreducible topological space.
Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x = 0 and y = 0, is not irreducible.