36 problems

Let $n$ be a positive integer, let $\mathbb{F}\subset\mathbb{C}$ be a field, and let $p_{1},\ldots,p_{n},q_{1},\ldots,q_{n}$ be fixed positive integers satisfying conditions C(i)--…

Derivation-module dimension conjecture. If

Let $L=L_\mathbf{0}\oplus L_\mathbf{1}$ be a quasireductive Lie superalgebra, and let an almost inner derivation of $L$ be a derivation that is almost inner, while an inner derivat…

Hussain–Ma–Yau–Zuo's conjecture. The graded algebra

Yan's conjecture. If $d$ is a simple derivation of $R_n$, then $\operatorname{Aut}(R_n)_d$ is conjugate in $\operatorname{Aut}(R_n)$ to a subgroup of the group of translations.

Let $k$ be a field of characteristic zero and let $A$ be a $k$-algebra of finite type. For each integer $q\ge 1$, let $Der^q(A)$ denote the module of differential operators of orde…

Let $n\geq 2$ be an integer. Let $(L,\cdot,[\cdot,\ldots,\cdot])$ be a transposed Poisson $n$-Lie algebra, and let $D$ be a derivation of both $(L,\cdot)$ and…

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ with $O_K=\mathbb{Z}[\zeta]$, and let $A$ be the matrix defined by … for $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in K$. Let $D$ be a…

The determinant-$p$ conjecture. The matrix $A=[a_{ij}]$ is a $(p-1)\times(p-1)$ matrix with determinant $p$.

Let $R$ be a Noetherian local ring with residue field $k$, and let $\operatorname{Der}_k(R)$ denote the $R$-module of $k$-derivations of $R$. Strong Zariski–Lipman conjecture. If ……

Let $R$ be a Noetherian local ring with residue field $k$, and let $\operatorname{Der}_k(R)$ denote the $R$-module of $k$-derivations of $R$. Herzog–Vasconcelos conjecture. If … th…

Let $K$ be a field of characteristic zero, let $q$ be a nonzero element of $K$ that is not a root of unity, and let $\mathcal{O}_{q}(G(k,n))$ be the quantum grassmannian with…

Divisibility conjecture. For any $m,i\geq 0$, the polynomial $\widetilde{f^m_i}(x,y)+\widetilde{f^m_i}(y,x)$ is divisible by

Let $A$ and $B$ be rings and let $\mathcal{T}$ be a triangular ring satisfying the following conditions: for $a\in A$, $aA=0$ implies $a=0$; and for $b\in B$, $bB=0$ implies $b=0$.…

Let $K$ be a field of characteristic $0$, let $\mathcal A$ be a $K$-algebra, and let $\delta$ be a locally nilpotent $K$-derivation or $K$-$\varepsilon$-derivation of $\mathcal A$.…

The LFED Conjecture. Images of locally finite $K$-derivations and $K$-$\mathcal{E}$-derivations of $K$-algebras are Mathieu subspaces.

Let $K$ be a field, let $n\geq 2$, and let $D$ be a simple derivation of the polynomial ring $K[x_1,\ldots,x_n]$. A $K$-subspace $M$ of a ring is a Mathieu-Zhao space if, whenever…

Higher-order derivation conjecture. If

Intersection-derivation conjecture. The set of derivations of $A$ is the union of the derivations of $A_1$ and $A_2$, restricted to $A$. This conjecture concerns how derivations be…

Let $A\subseteq\mathbb{K}[x]$ be a finite-codimension subalgebra, let $k_\alpha$ be the dimension of the vector space of all $\alpha$-derivations of $A$, and let…

Let $A\subseteq\mathbb{K}[x]$ be a subalgebra with spectrum containing $\alpha$, let $C=\{\alpha_1,\ldots,\alpha_s\}$ be its spectrum, and let $\alpha_j\sim\alpha$ mean that…

Halperin's conjecture. If $H^*$ is a positively elliptic algebra, then $H^*$ does not admit a non-trivial derivation of negative degree.

Let $R$ be a local $k$-algebra such that the $R$-module of $k$-derivations $\operatorname{Der}_k(R)$ is finitely generated. Write $\operatorname{Gdim}$ for Gorenstein dimension and…

Let $K$ be a field of characteristic zero, let $\mathcal{A}$ be a $K$-algebra, and let $\delta$ be a locally finite derivation or E-derivation of $\mathcal{A}$. A $K$-subspace $M$…

LFED conjecture. Let $K$ be a field of characteristic zero and $\mathcal{A}$ a $K$-algebra. Then for every locally finite derivation or $\mathcal{E}$-derivation $\delta$ of…