On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

Abstract

We study the $ZG$-module $A$ such that $Z$ is the ring of integers, the group $G$ has infinite section $p$-rank (or infinite 0-rank), $C_G(A) = 1$, $A$ is not a minimax $Z$-module, and, for every proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank, respectively), the quotient module $A/C_A(H)$ is a minimax $Z$-module. It is proved that if the group $G$ under consideration is locally solvable, then $G$ is a solvable group. Some properties of a solvable group of this type are obtained.