80-year-old Kaplansky's Conjecture is broken by Giles Gardam

in MES Science2 months ago (edited)

There are known 3 Kaplansky Conjecture related with group rings KG for K - a field and G - torsion-free group.

  1. KG does not contain nontrivial units (with support consisting of more than 1 element)

  2. 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
https://www.quantamagazine.org/mathematician-disproves-group-algebra-unit-conjecture-20210412/

And here is his full paper:
https://arxiv.org/abs/2102.11818v3

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.

Best wishes.