On groups that are residually of finite rank

Let r be a fixed positive integer. A group G has (Prufer) rank r if every finitely generated subgroup of G can be generated by r elements and r is the least such integer. In this paper we consider groups that are residually of rank r. Among other things we prove that a periodic group that is residually (of rank r and locally finite) is locally finite and obtain the structure of groups that are residually (of rank r and locally soluble). A number of examples are also given to illustrate the limitations of these theorems.