3 problems

Goodearl's conjecture. If $R$ is an affine algebra of GK-dimension $2$, then its Jacobson radical $J(R)$ is nil.

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…

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