There are known 3 Kaplansky Conjecture related with group rings KG for K - a field and G - torsion-free group.
KG does not contain nontrivial units (with support consisting of more than 1 element)
KG does not contain nonzero idempotents
3 KG does not contain nonzero zero divisors
Giles Gardam disproved the first one, here is online talk when he revealed that fact:
Here is more soft for non-mathematicians article about it
And here is his full paper:
When I attended course in Group rings last year with Professor Zbigniew Marciniak, he stated these conjectures as very very nice and important.
A few weeks ago at Algebra seminar I gave a talk in which I proved Connell's Theorem stating that KG is prime iff G is torsion-free group (theorem from 1963).
I like that he disproved conjecture by giving concrete counterexample.