On embedding of group rings of residually torsion free nilpotent groups into skew fields

Eizenbud, A.; Lichtman, A. I.

doi:10.1090/S0002-9947-1987-0869417-3

On embedding of group rings of residually torsion free nilpotent groups into skew fields
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by A. Eizenbud and A. I. Lichtman PDF

Trans. Amer. Math. Soc. 299 (1987), 373-386 Request permission

Abstract:

It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin’s theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring $K\left \langle G \right \rangle$ of a residually torsion free nilpotent group $G$. It is shown that its subfield $D$, generated by the group ring $KG$, does not depend on the order of $G$ for many kinds of orders and the study of $D$ can be reduced in some sense to the case when $G$ is nilpotent.

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The theory of groups