# Some results relevant to embeddability of rings (especially group algebras) in division rings

## TL;DR

This paper explores conditions under which rings, especially group algebras, can be embedded into division rings, building on Cohn's matrix-based characterization and examining module-induced data to address longstanding embedding questions.

## Contribution

It develops a module-based framework to analyze embeddability of rings into division rings, extending Cohn's matrix conditions and applying this to group algebras.

## Key findings

- Re-proves a partial embedding result by Dubrovin.
- Identifies conditions for modules to induce embeddability data.
- Suggests potential generalizations for embedding group algebras.

## Abstract

P. M. Cohn showed in 1971 that given a ring $R$, to describe, up to isomorphism, a division ring $D$ generated by a homomorphic image of $R$ is equivalent to specifying the set of square matrices over $R$ which map to singular matrices over $D,$ and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free $R$-modules. In this note, we examine the latter concept further, and show how an $R$-module $M$ satisfying certain conditions can be made to induce such data. In an appendix we make some observations on Cohn's original construction, and note how the data it uses can similarly be induced by appropriate sorts of $R$-modules. Our motivation is the longstanding question of whether, for $G$ a right-orderable group and $k$ a field, the group algebra $kG$ must be embeddable in a division ring. Our hope is that the right $kG$-module $M=k((G))$ might induce a closure operator of the required sort. We re-prove a partial result in this direction due to N. I. Dubrovin, note a plausible generalization thereof which would give the desired embedding, and briefly sketch some thoughts on other ways of approaching the problem.