Department of Mathematics

Given a group \( G \) and a field \( K \), the group ring \( K[G] \) is the set of finite formal sums of elements of the group with coefficients in \( K \).

Kaplansky conjectured that if \( G \) is torsion free, then \( K[G] \) should not contain units, zero divisors, or idempotents. In 2021, Giles Gardam showed the units conjecture to be false, though the other two conjectures are still wide open.

In this talk, I will review the Kaplansky conjectures, and discuss algebraic properties of a group which guarantee that it is torsion-free, and which simultaneously guarantee that the group satisfies the zero divisor conjecture. My goal for the talk will be to explain the background behind two open questions that I have spent considerable time thinking about over the last decade, but which seem quite difficult—in the sense that I have little to show for my efforts!