The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras.
Facts
Partially Attested
StatementThe group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. 1 This entity groups several distinct conjectures by Kaplansky (group ring conjectures, a Banach algebra conjecture, and a quadratic forms conjecture); the value given here describes the group ring zero divisor conjecture, one of the set. Proposed YearThis year is documented for the quadratic forms (u-invariant) conjecture specifically; the other conjectures grouped under this entity do not have a documented single proposal year in this source. Progress Toward ResolutionThe unit conjecture was disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a crystallographic group. 1 Different sub-conjectures have different fates: the unit conjecture was disproved in 2021, the quadratic forms conjecture was refuted in 1989, and the zero divisor and idempotent conjectures remain open. Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Kaplansky's Conjectures (Wikipedia)
Group ring conjectures section, zero divisor conjecture
The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain.
Quadratic forms conjecture section
In 1953, Kaplansky proposed the conjecture that finite values of u-invariants can only be powers of 2.
Unit conjecture section
The unit conjecture, however, was disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a crystallographic group.
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.