18 problems

Let $G_n(\mathbb{C})$ denote the set of pairs $(A,B)$ of $n\times n$ complex matrices that generate the matrix ring, and let $\operatorname{msl}_{\mathbb{C}}(A,B)$ be their matrix…

Let $S$ be a commutative ring and let $\mathbb{M}_2(S)$ denote the ring of $2\times 2$ matrices over $S$. An element is left strongly regular if it satisfies $a=a^2x$ for some $x$…

Nonconverse conjecture. Additive periodicity of ${\rm M}_n(R)$ does not imply additive periodicity of $R$.

Let $n$ be a positive integer and let $F$ be a finite field. Consider the ring of $n\times n$ matrices over $F$, together with its commuting graph, whose vertices are the non-centr…

Let $F$ be a field, let $M_n(F)$ be a matrix ring over $F$, and let $p$ be a multilinear polynomial. The image $p(M_n(F))$ consists of all values obtained by evaluating $p$ on…

Larsen's conjecture. For a given $k$, there exists a constant $N(k)\in\mathbb N$ depending only on $k$ such that, for all pairs $(n,q)$, if

Weakly periodic division-ring conjecture. The matrix ring $\mathbb{M}_n(D)$ is weakly periodic if and only if $D$ is a finite field.

Periodic matrix-ring conjecture. If $R$ is a periodic semi-perfect ring, then the matrix ring

Let $n$ be a positive integer, and let $\mathbb{M}_{n}(\mathbb{F}_{2})$ denote the ring of $n\times n$ matrices over the field with two elements. An idempotent is an element $e$ sa…

Let $m,n\geq 2$ be natural numbers and let $p$ be a prime. Consider the matrix ring $\mathbb{M}_n(\mathbb{Z}_{p^m})$, where $\mathbb{Z}_{p^m}$ is the ring of integers modulo $p^m$.…

Determinant-sum conjecture. There is a sufficiently large constant $C>1$ such that, for arbitrary subsets $E,F\subseteq M_2(\mathbb F_q)$ satisfying

Let $d>1$ and let $R$ be a finite commutative ring. For $k\geq 1$, write $S_k(\mathbb{M}_d(R))=\sum_{A\in\mathbb{M}_d(R)}A^k$. The unique element $e\in R\setminus\\{0\\}$ such that…

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 $p$ be a prime, let $s_1\geq s_2\geq\cdots\geq s_r$, and let $M_{p^{s_i}}^d$ denote the ring of $d\times d$ matrices over $\mathbb{Z}/p^{s_i}\mathbb{Z}$. For an $r$-tuple…

Power-sum conjecture. The sum $S_k^d(R)$ is $0$ unless $d=2$, $\operatorname{card}(R)\equiv 2\pmod{4}$, $1<k\equiv -1,0,1\pmod{6}$, and the unique element $e\in R\setminus\{0\}$ sa…

Diameter-five conjecture. The diameter of $\Gamma(\mathrm{M}_n(\mathbb{R}))$ is at most $5$.

Let $K$ be a field, let $n,d\in\mathbb{N}$, and write $L_K(1,n)$ for the Leavitt algebra of type $(1,n)$. The notation $\operatorname{M}_d(L_K(1,n))$ denotes the ring of…

Let $R$ be a ring with identity, let $\mathds F$ be a finite field, and let $\Gamma(R)$ denote the commuting graph of $R$, whose vertices are the non-central elements and whose edg…