4 problems

Let $R=\bigoplus_{n\in G}R_n$ be a ring graded by a torsion-free group $G$, and let $J(R)$ denote its Jacobson radical. An ideal is graded when it is the direct sum of its homogene…

Let $R$ be a ring with property (T), characteristic $p$, and order $p^{nr}$, whose maximal Galois subfield is $F={\bf F}_p$. Fix the invariants $p$, $n$, $r$, $s$, $t$, and…

Let $A$ be a profinite algebra that is left semiartinian, equivalently semilocal with Jacobson radical $J$ right T-nilpotent. The semiartinian profinite algebra conjecture. The Jac…

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