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

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 ⟨ G ⟩ K\left \langle G \right \rangle of a residually torsion free nilpotent group G G . It is shown that its subfield D D , generated by the group ring K G KG , does not depend on the order of G G for many kinds of orders and the study of D D can be reduced in some sense to the case when G G is nilpotent.