41 problems

Let $D$ be a division ring, and write $D^*$ for its multiplicative group of nonzero elements. Lichtman's conjecture. If $D$ is noncommutative, then $D^*$ contains a non-cyclic free…

Herstein's conjecture. In any characteristic, every subgroup of $D^\times$ of odd order is cyclic.

Let $D$ be a division ring, and let $H$ be a subnormal subgroup of the multiplicative group $D^\times$ of $D$. Gonçalves–Mandel conjecture. The group $H$ contains a non-cyclic free…

Let $G$ be a torsion-free group and let $\mathbb{K}$ be a field. The division-ring embedding conjecture. The group algebra $\mathbb{K}[G]$ embeds into a division ring. This is a st…

Let $g>0$. A simple tiling $\mathcal T$ of a genus-$g$ surface has an associated incidence theorem over a division ring ${\mathrm D}$, and $n$ denotes the dimension of the ambient…

Let $\mathcal T$ be a simple tiling of a surface of positive genus, meaning that its underlying graph is $(n+1)$-degenerate, and let ${\mathrm D}$ be a division ring. The incidence…

Let $G$ be a finitely presented group and let $F$ be a field. Denote by $b^{(2)}_1(G;\mathcal N(G))$ the first $L^2$-Betti number with coefficients in the group von Neumann algebra…

A group $G$ is a Lewin group if, for every field $F$ with a $G$-action, the crossed product $F\ast G$ admits a Hughes-free $F\ast G$-division ring that is its universal division ri…

A Linnell group is a group for which, for every field $F$ and crossed product $F\ast G$, a Linnell division ring $\iota\colon F\ast G\to\mathcal D_{F\ast G}$ exists and is unique u…

Let $G$ be a torsion-free group and let $kG$ be its group algebra over a field $k$. A division ring has the Linnell property when it has the property referred to in the source. Jai…

Additive-map factorization conjecture. There exists a fixed element $q\in D$ such that

Universal Hughes-free embedding conjecture. Group algebras of all locally indicable groups have Hughes-free embeddings that are universal in a suitable sense.

Kaplansky's zerodivisor conjecture. The group algebra $kG$ is a domain.

Let $K$ be a field and let $G$ be a torsion-free group. A Linnell division $K[G]$-ring is an epic division $K[G]$-ring that is $(N,G)$-free for every subgroup $N$ of $G$. Linnell d…

Let $K$ be a field and let $G$ be a locally indicable group. A Hughes-free division $K[G]$-ring is an epic division $K[G]$-ring that is $(N,H)$-free for every pair of subgroups…

Strict containment conjecture. The class of weakly locally finite division rings strictly contains the class of Stewart's division rings.

Let $D$ be a division ring with center $F$ and multiplicative group $D^*$. A subgroup $N$ of $D^*$ is subnormal if it admits a finite chain of subgroups from $N$ to $D^*$ in which…

Let $n\in\mathbb{N}$ with $n>1$, and let $\varphi:F\to M_n(G)$ be a ring embedding, where $G$ and $F$ are division rings. Suppose that $a$ and $b$ are positive integers satisfying…

Let $D$ be a non-commutative division ring algebraic over its center. A division subring is centrally finite if it has finite dimension over its center. Mahdavi-Hezavehi's weak Kur…

Let $D$ be a division ring algebraic over its center $F$. A division subring is finitely generated if it is generated by finitely many elements, and $D$ is locally finite if every…

Let $D$ be a division ring with center $F$. A subfield $K$ of $D$ is maximal if it is maximal among subfields of $D$. Maximal-subfields algebraicity conjecture. If every maximal su…

Let $D$ be a division ring and let $D^*$ be its multiplicative group. A subgroup is non-central if it is not contained in the center of $D$. Weak Herstein–Scott conjecture. Every n…

Let $D$ be a division ring and let $D^*$ denote its multiplicative group. A subgroup of $D^*$ is subnormal if it is connected to $D^*$ by a finite chain of normal subgroup inclusio…

Let $D$ be a skew field with center $Z$. Say that a noncommutative free group $Z$-algebra is a $Z$-algebra generated by a noncommutative free group. Free group algebra conjecture.…

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…