The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any characteristic ?
Does the conjecture imply that any finitely generated projective $F[G]$-module is free ? or eventually stably free?
For which field $F$ and which class of groups $G$, the conjecture is known to be true?
Kaplansky's zero divisor conjecture: Let $\mathbb{F}$ be a field and $G$ be a torsion-free group. Then $\mathbb{F}[G]$ does not contain a zero divisor.
The existence of a nontrivial idempotent $a$ in a ring $R$ implies the existence of a zero divisor in $R$, because $a(a-1)=0$. So, the zero divisor conjecture implies the idempotent conjecture.
The idempotent conjecture has been confirmed in special cases. For example, Formanek (1973) showed that if $G$ is a torsion-free group satisfying the ascending chain condition on cyclic subgroups and $\mathbb{F}$ is a field of characteristic $0$, then $\mathbb{F}[G]$ has no nontrivial idempotents. Also, Bass (1976) proved that if $G$ is a torsion-free linear group, then $\mathbb{C}[G]$ has no nontrivial idempotents.
These conjectures have not been confirmed for any fixed field and it seems that confirming the conjecture even for the finite field $\mathbb{F}_2$ is still out of reach.
For the zero divisor conjecture case, our work Zero divisors and units with small supports in group algebras of torsion-free groups may be helpful. Also for more details about these two conjectures you can see Zero-divisors and idempotents in group rings.
