4 problems

Null ideal conjecture. A small profinite ring has an open null ideal.

Nilpotence conjecture. The Jacobson radical $J(R)$ is nilpotent. In particular, $R$ has an open nilpotent ideal.

Let (A), (B), (C) denote the three intermediate conjectures for small profinite groups, and let (B') denote the ring conjecture that nilpotent-by-finite small profinite rings are n…

Let $(R,G)$ be a small profinite ring. Intermediate conjectures for small profinite rings. The following assertions should hold: (A') $R$ is nilpotent-by-finite; and (B') if $R$ is…