Zero-divisors and idempotents in group rings

Abstract: After a brief introduction of the basic properties of group rings, some famous theorems on traces of idempotent elements of group rings will be presented. Next we consider some famous conjectures stated by Irving Kaplansky, among them the zero-divisor conjecture. The conjecture asserts that if a group ring is constructed from a field (or an integral domain) and a torsion-free group, then it does not contain any non-trivial zero-divisors. Here we show how a confirmation of the conjecture for certain fields implies its validity for other fields.