A finite group \(G\) is said to have (special) rank \(r=\text{rk}(G)\) if every subgroup of \(G\) can be generated by \(r\) elements, and \(r\) is minimal with this property. If \(p\) is a prime, the \(p\)-rank \(\text{r}_p(G)\) of a finite group \(G\) is the rank of a Sylow \(p\)-subgroup of \(G\). Let now \(G\) be an arbitrary group, \(\{H_i\mid i\in I\}\) the family of all finite factor-groups of \(G\). The upper rank \(\text{ur}(G)\) of \(G\) is \(\sup\{\text{rk}(H_i)\mid i\in I\}\), and respectively, the upper \(p\)-rank \(\text{ur}_p(G)\) of \(G\) is \(\sup\{\text{r}_p(H_i)\mid i\in I\}\). The author considers the following Conjecture A. Let \(G\) be a finitely generated soluble group. If \(\text{ur}_p(G)\) is finite for every prime \(p\) then \(G\) has finite upper rank. The study of the soluble groups is allied to the study of modules over integral group rings. Let \(Q\) be a group, \(R=\mathbb{Z} Q\) the integral group ring of \(Q\), \(M\) an \(R\)-module. For a prime \(p\) denote by \(\{B_j\mid j\in J\}\) the set of all finite \(R\)-factor-modules of \(M/pM\), and put \(\text{ur}_p(M)=\sup\{\dim_{F_p}(B_j)\mid j\in J\}\) and \(\text{ur}(M)=\sup\{\text{ur}_p(M)\mid p\) runs over all primes\}. Let \(Q\) be a minimax group, \(M\) a \(\mathbb{Z} Q\)-module. Then \(M\) is called quasi-finitely generated if there exists a finitely generated group which is an extension of \(M\) by \(Q\). Conjecture A is equivalent to Conjecture B. Let \(Q\) be a finitely generated minimax group and \(M\) a quasi-finitely generated \(\mathbb{Z} Q\)-module. If \(\text{ur}_p(M)\) is finite for every prime \(p\) then \(M\) has finite upper rank. Theorem 3.1. Let \(Q\) be a minimax group which is Abelian-by-polycyclic, \(K\) a finitely generated commutative ring and let \(M\) be a finitely generated \(KQ\)-module. If \(\text{ur}_p(M)\) is finite for every prime \(p\) then \(\text{ur}(M)\) is finite. -- Corollary. The statement of conjecture A holds for every finitely generated group \(G\) which is the semi-direct product of an Abelian group by an Abelian-by-polycyclic minimax group. The main result of this paper is Theorem 3.8. Let \(Q\) be a minimax group which is Abelian-by-polycyclic, \(K\) a finitely generated commutative ring and let \(M\) be a finitely generated \(KQ\)-module. Suppose that there is a finite set \(\sigma\) of primes such that \(\text{ur}_p(M)\) is finite for all \(p\in\sigma'\). Then (i) there is \(t\in\mathbb{N}\) such that \(\text{ur}_p(M)\leq t\) for all \(p\in\sigma'\); (ii) \(\pi(M)\) is finite; (iii) there is \(s\in\mathbb{N}\) and a finite set \(\phi\) of primes such that \(\text{ur}_p(A)\leq s\) for all \(p\in\phi'\) and every \(KQ\)-submodule \(A\) of \(M\).