In the representation theory of finitely generated nilpotent groups the use of induction leads one to consider impervious modules, those modules contain no non-zero submodules induced from a module over a subgroup of infinite index. Let \(H\) be a finitely generated nilpotent group with centre \(Z\) and let \(R\) be a commutative ring. Let \(H_0\) be a normal subgroup of \(H\) containing \(H'Z\) and with \(H_0/H'Z\) finite-by-cyclic. If \(M\) is an impervious \(RH\)-module of the group ring \(RH\) which is \(RZ/P\)-torsion-free for some prime ideal \(P\) of \(RZ\) with \((1+P)\cap Z=1\), then the authors prove that \(M\) is \(RH_0/PRH_0\)-torsion-free. This is the main result of this paper. Moreover, an impervious \(RH\)-module torsion-free over \(RZ\) is also torsion-free as an \(R(H'Z)\)-module. If \(R\) is the ring of integers and \(H\) is non-Abelian, then \(k\)-\(\dim(M)\geq h(H'Z)-h(Z)+2\). Here \(k\)-dim is the Krull dimension and \(h\) denotes the Hirsch length.
Let \(G\) be a weakly finitely presented Abelian-by-nilpotent group with Fitting subgroup \(F\). Suppose that some subgroup of \(G/F\) of finite index is a \(d\)-generator group. Then, applying the results above, the authors prove that some subgroup of \(G/F\) of finite index is of class at most \(d\).