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The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. __NOTOC__ ==Kaplansky's conjectures on groups rings== Let ''K'' be a field, and ''G'' a torsion-free group. Kaplansky's zero divisor conjecture states that the group ring ''K''() does not contain any zero divisors, that is, it is a domain. No counterexamples have been found and the conjecture has been proved for wide classes of groups. Two related conjectures are: * ''K''() does not contain any non-trivial units - if ''ab'' = ''1'' in ''K''(), then ''a'' = ''k.g'' for some ''k'' in ''K'' and ''g'' in ''G''. * ''K''() does not contain any non-trivial idempotents - if ''a2'' = ''a'', then ''a'' = ''1'' or ''a'' = ''0''. The zero-divisor conjecture implies the idempotent conjecture and is implied by the units conjecture. As of 2014 all three are open. Another related conjecture is the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture, which generalises Kaplansky's idempotent conjecture to the reduced group C *-algebra. The idempotent and zero-divisor conjectures have implications for geometric group theory. If the Farrell-Jones conjecture holds for ''K''() then so does the idempotent conjecture. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kaplansky's conjecture」の詳細全文を読む スポンサード リンク
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