Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will discuss these conjectures and their relationship to other conjectures and properties of groups. I will then explain how modern solvers for Boolean satisfiability can be applied to them, producing the first counterexample to the unit conjecture.