Kourovka Notebook Problem 1.6

Definitions and mathematical statement

A group $G$ is right-orderable if it admits a strict total order $<$ such that \[ \forall g,h,x\in G,\qquad g<h\Longrightarrow gx<hx. \] For a specified nonzero associative ring $R$ with identity, the group ring $R[G]$ consists of finite formal sums $\sum_{g\in G}r_g g$, with coefficientwise addition and multiplication determined by \[ (rg)(sh)=(rs)(gh)\qquad(r,s\in R,\ g,h\in G). \] A skew field is a nonzero associative ring with identity in which every nonzero element has a multiplicative inverse. Under the unital convention, an embedding into a skew field is an injective ring homomorphism preserving the identity.

The supplied question asks whether the group ring of every right-orderable group embeds into a skew field. Once an admissible coefficient ring $R$ is specified, its formulation is \[ \forall G\;\left(G\text{ is right-orderable}\Longrightarrow \exists D\,\exists\iota:R[G]\hookrightarrow D\; \left(D\text{ is a skew field and }\iota\text{ is a unital ring homomorphism}\right)\right). \] The source does not specify the coefficient ring, its hypotheses, or its quantification. Thus this display is a formulation scheme for a specified $R$, not a resolution of the intended coefficient domain. The unital embedding convention is also not explicit in the source.

Short English statement

Does the group ring of every right-orderable group admit an injective homomorphism into a skew field? The supplied statement leaves the coefficient ring and its quantification unspecified, so a fully determined mathematical statement requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $700006$, source code catalogue entry, title Kourovka Notebook Problem 1.6; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied reference: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 1.6, first issue (1965); cited edition: 21st issue (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. No external source was independently consulted.
• Attribution: the statement prefixes the question with A. I. Mal'cev; the background and proposer field name L. A. Bokut', with proposed year 1965. The supplied record does not explain the relationship between these attributions.
• The registry labels this problem open. This is supplied source metadata, not an independently verified assessment of its mathematical status.