The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group, then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
Giles Gardam. "A counterexample to the unit conjecture for group rings." Ann. of Math. (2) 194 (3) 967 - 979, November 2021. https://doi.org/10.4007/annals.2021.194.3.9