Kourovka Notebook Problem 1.5

Definitions and mathematical statement

Let $R$ be a nonzero associative ring with identity and let $G$ be a group. The group ring $R[G]$ consists of formal sums $\sum_{g\in G}r_g g$, where $r_g\in R$ and only finitely many coefficients are nonzero. Addition is coefficientwise and multiplication is determined by $(rg)(sh)=(rs)(gh)$. A nonzero ring $A$ has no zero divisors if \[ \forall a,b\in A,\qquad ab=0\Longrightarrow(a=0\ \text{or}\ b=0). \] A skew field is a nonzero associative ring with identity in which every nonzero element is invertible. An embedding of rings is taken here to be an injective identity-preserving ring homomorphism.

For a specified coefficient ring $R$, the question has the following precise form: does there exist a group $G$ such that $R[G]$ has no zero divisors and, for every skew field $D$, there is no injective identity-preserving ring homomorphism $R[G]\to D$?

The supplied source asks whether there exists a group whose group ring has these properties, but does not specify the coefficient ring or its quantification. Consequently, $R$ remains an unresolved parameter: the displayed formulation is a conditional clarification, and does not determine whether the intended question fixes particular coefficients or allows their existential choice. The source describes this as a "well-known problem."

Short English statement

The problem asks for a group whose group ring has no zero divisors but cannot be embedded in any skew field. The coefficient ring is unspecified in the supplied statement, so the intended existence question requires clarification. The supplied registry labels the problem open; this status has not been independently verified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700005$, exact source code catalogue entry, title Kourovka Notebook Problem 1.5. The record supplies the status open and an update timestamp of 2026-10-16T14:00:00Z; these are registry metadata, not independent mathematical or literature verification.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 1.5, first issue (1965); the supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by L. A. Bokut', with proposed year 1965. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This reference and attribution are supplied source information and have not been independently checked.