PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 104, Number 3, November 1988 APPLICATIONS OF A NEW AT-THEORETIC THEOREM TO SOLUBLE GROUP RINGS P. H. KROPHOLLER, P. A. LINNELL AND J. A. MOODY (Communicated by Donald S. Passman) ABSTRACT. Let R be a ring and let G be a soluble group. In this situation we shall give necessary and sufficient conditions for RG to have a right Artinian right quotient ring. In the course of this work, we shall also consider the Goldie rank problem for soluble groups and record an affirmative answer to the zero divisor conjecture for soluble groups. 1. Introduction. We shall start by describing the theorem referred to in the title. Let fibea ring (all rings in this paper will have a one), let G be a group, and let R * G denote a crossed product (see [11]). Thus R*G may be viewed as a free i?-module with basis {g\g 6 G} where each g is a unit in R * G, and ring structure satisfying RgRh = Rgh. Of course, R * G is not uniquely determined by R and G. However if H < G, then R * H is defined to be (&heH Rh, a subring of R * G. Another way of describing R * G is that it is a G-graded ring (or group-graded ring) with a unit in each degree [5, §5]. For a right Noetherian ring S, let Go{S) denote the Grothendieck group as- sociated with the category of all finitely generated right S-modules. If R is right Noetherian and G is polycyclic-by-finite, then R*G is also right Noetherian. More- over if H < G, then the functor M i—> M <8>r*h R * G preserves exact sequences and hence induces a natural group homomorphism G0(R * H) —► Go(R * G) which is called the induction map. The following theorem was proved by the third author in his thesis [7] and described in [8] (see [9] for full details). THEOREM l. I. Let R be a right Noetherian ring, let G be a polycyclic-by-finite group, and let &~(G) denote the set of finite subgroups of G. Then the natural (induction) map ©F&y-(G\ Go(R * F) —+ Go(R * G) is surjective. It is well known that such a theorem would have important consequences in algebra, and the purpose of this paper is to consider a few of these in the theory of group rings of soluble groups. Ever since the papers of Brown [2] and Farkas and Snider [6], it has been folklore that the zero divisor conjecture for soluble-by-finite groups (cf. Theorem 1.4) would follow from Theorem 1.1, but this fact has never been published because it was never expected that such a theorem would be proved. An indication of how to deduce Received by the editors April 20, 1987 and, in revised form, October 26, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A08, 16A27; Secondary 19M05. Key words and phrases. Group ring, zero divisor conjecture, Goldie rank, quotient ring, Grothendieck group. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 675 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use