Introduction

The study of group rings of soluble groups of finite rank and modules over them was begun with famous papers of P. Hall, where he obtained the first results on this subject. Some later these investigations were developed in very important papers of J. Roseblade. These results were followed by many interesting papers on polycyclic group rings and modules over them by other investigators and the list of these papers is too large to be presented here.

Some theorems proved for modules over polycyclic group rings are formulated more naturally for soluble groups of finite rank. In fact, the real cause of some properties of group rings of polycyclic groups is finiteness of rank of these groups. For this and many other reasons, most notably due to the needs of the theory of soluble groups, group rings of soluble groups of finite rank attract attention of many investigators. We must note in particular the following papers of Brookes, Brown, Musson, Wehrfritz, Wilson, Zaitsev. But we should note that theory of group rings of soluble groups of finite rank and modules over them is not so deep and completed as in the “polycyclic” case. In this survey we consider some techniques, approaches and conjectures which might be useful for further progress in this domain.