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The present paper is devoted to the study of finitely generated modules over nilpotent groups of finite rank and some problems connected with them. In it the approach suggested in [I] to the investigation of modules of this kind is developed. The essence of this approach is the local application of the methods and results of Hall [2, 3] concerning finitely generated solvable groups.Let A be a finitely generated ~G-module, ~ being a principal ideal ring, ~ being a locally almost polycyclic group of finite free rank. We shall denote by ~(~# the complete set of (nonassociated) As a rule, as the group ~ we consider a nilpotent group of finite free rank, and as either the ring Z , or ~<~> , the group algebra of the infinite cyclic group <~ over a finite field~. As is known, these two kinds of rings are the most important for applications.