Kourovka Notebook Problem 11.122

Definitions and mathematical statement

A group $G$ is torsion-free if $g^n=1$, for $g\in G$ and an integer $n\geq 1$, implies $g=1$. For a specified coefficient ring $A$ with identity, let $R=A[G]$ denote its group ring: the ring of finite formal sums $\sum_{g\in G}a_g g$, with multiplication determined by $(ag)(bh)=(ab)(gh)$.

Using left modules as a convention, a free $R$-module is a module isomorphic to $\bigoplus_{i\in I}R$ for some set $I$, with no finiteness restriction on $I$. A free cyclic submodule of a module $M$ is a submodule $Rm$, generated by an element $m\in M$, for which the map $R\to Rm$, $r\mapsto rm$, is an isomorphism. Equivalently, $\{r\in R:rm=0\}=\{0\}$.

For the coefficient ring intended by the source, the question is whether the following assertion holds: for every torsion-free group $G$, every set $I$, and every nonzero left $A[G]$-submodule $M\leq\bigoplus_{i\in I}A[G]$, there exists $m\in M\setminus\{0\}$ such that \[ \forall r\in A[G],\qquad rm=0\ \Longrightarrow\ r=0. \] The supplied statement does not specify the coefficient ring or quantify over coefficient rings. Thus $A$ in this formulation is an unresolved parameter, not an additional universal quantifier. If the intended group ring is the integral group ring, the displayed assertion is to be read with $A={\mathbb{Z}}$; this interpretation requires confirmation. The source also leaves module sidedness unstated. For right modules, the corresponding formulation uses $mR$ and the condition $mr=0\Longrightarrow r=0$.

Short English statement

The question asks whether every nonzero submodule of a free module over the group ring of a torsion-free group contains a submodule isomorphic to that group ring as a module over itself. The coefficient ring and module sidedness are unspecified in the supplied source.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700814$, exact source problem code catalogue entry, title Kourovka Notebook Problem 11.122. The registry labels the problem open; this is supplied metadata, not an independently verified assessment.
• Attributed to A. L. Shmel'kin, proposed in 1990. The supplied background cites The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.122, and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied registry record was created and updated at 2026-10-16T14:00:00Z, with upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. These are source-record timestamps, not independent literature-verification dates. No external source was consulted for this draft.