10 problems

Let $R$ be a Noetherian ring, complete with respect to a positive filtration topology, and let $(\sigma,\delta)$ be a continuous skew derivation compatible with the filtration. Def…

Let $\mathfrak{g}=sl_2(\Bbbk)$ and let $A=A_n$ be as in Theorem. Let $\mathcal{F}$ be an object of ${\bf Alg}_1$, and let $G_{\mathfrak{g},A}(\mathcal{F})$ and…

Let $A$ be an almost commutative separated local ring. Denote by $\bar{A}$ the intersection of all local Henselian rings $H$ in the completion $\hat{A}$ such that…

Semiprimeness conjecture. The ring $Q^+[[x;sigma,delta]]$ is (semi)prime.

Bavula's quotient-ring conjecture. There is an isomorphism

Let $R$ be a one-sided Archimedean ring, meaning that $R$ is either right Archimedean or left Archimedean. Reduced one-sided Archimedean ring conjecture. The ring $R$ is reduced if…

Let $L/K$ be an extension of subfields of $\mathbb{C}$, and let $G$ be a group satisfying the strong Atiyah conjecture. Let $\mathcal D_{K[G]}$ and $\mathcal D_{L[G]}$ be, respecti…

Non-commutative power-sum conjecture. The theorem for finite commutative rings remains true for non-commutative $R$: $S_k^d(R)=0$ unless $d=2$,…

Let $K$ be a field of zero characteristic, let $A_n=K\langle x_1,\ldots,x_n,\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n}\rangle$ be the Weyl algebra, and let…

Let $M$ be a square matrix of size $m$ whose entries are functions from a set $S$ to a reduced noncommutative ring $R$. For $v\in S$, write $M|_v$ for the matrix obtained by evalua…