Kourovka Notebook Problem 16.23

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ with $n>2$, and let $A$ be a ring with identity. For the usual associative-ring interpretation, write $\mathrm{GL}_n(A)$ for the group of invertible matrices and define the elementary subgroup by \[ E_n(A)=\langle I_n+a e_{ij}:a\in A,\ 1\leq i,j\leq n,\ i\neq j\rangle\leq\mathrm{GL}_n(A), \] where $e_{ij}$ is the matrix unit. For each proper two-sided ideal $J\subsetneq A$, let $\rho_J:M_n(A)\to M_n(A/J)$ be entrywise reduction. Interpreting a scalar matrix as any matrix $\lambda I_n$ with $\lambda\in A/J$, the required nonscalarity condition is \[ \forall J\subsetneq A\text{ a two-sided ideal},\quad \forall\lambda\in A/J,\qquad \rho_J(g)\neq\lambda I_n. \] For a group $H$, put \[ \operatorname{Comm}(H)=\{xyx^{-1}y^{-1}:x,y\in H\}. \] The supplied question asks whether there exist $n>2$, a ring $A$ with identity, and $g\in E_n(A)$ satisfying the displayed nonscalarity condition such that $g$ is not a single commutator. The ambient group for the commutator is not specified. For an explicitly chosen ambient group $H$ containing $E_n(A)$, the question becomes \[ \exists n\in{\mathbb{N}},\ n>2,\ \exists A,\ \exists g\in E_n(A):\quad \left(\forall J\subsetneq A\text{ a two-sided ideal},\ \forall\lambda\in A/J,\ \rho_J(g)\neq\lambda I_n\right) \ \land\ g\notin\operatorname{Comm}(H). \] In particular, taking $H=E_n(A)$ and taking $H=\mathrm{GL}_n(A)$ give distinct formulations; the supplied record does not identify which is intended. Associativity, the two-sided interpretation of ideals, and the meaning of scalar matrices over a possibly noncommutative ring also require confirmation from the source conventions. No commutativity assumption on $A$ is supplied.

Short English statement

L. Vaserstein asks whether some elementary matrix of size greater than two remains nonscalar in every proper quotient ring yet cannot be expressed as a single commutator. The intended ambient group for commutators and the ring conventions require clarification. The registry labels the problem open; this is a supplied status, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $701215$, exact source problem code catalogue entry, title Kourovka Notebook Problem 16.23. The supplied record labels the problem open and attributes it to L. Vaserstein, proposed in 2006.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), Problem 16.23; also identified in the record with the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.