The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding W ∗ {W^ * } group algebra is exploited to prove that certain localizations of a group ring K G KG possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain K G KG -modules have zero Euler characteristic. The assumptions are that K K is a commutative integral domain of characteristic zero, and that G G has a nontrivial, torsion-free, abelian normal subgroup A A . The main result of the paper is that the localization of K G KG obtained by inverting all elements of the form α − a \alpha - a , where α \alpha is a nonzero element of K K and a a is a nontrivial element of A A , has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of K G KG obtained by inverting all nonzero elements of K A KA .) Given a K G KG -module M M which has a finite free resolution, such that M M is finitely generated over K K , it is proved that the Euler characteristic of M M is zero. This verifies an unpublished result of Rosset.