Relationships between Almost Completely Decomposable Abelian Groups and Their Multiplication Groups

Let [Formula: see text] be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, we study relationships between the above groups [Formula: see text] and their multiplication groups Mult [Formula: see text]. It is proved that groups from [Formula: see text] are determined by their multiplication groups. For a rigid group [Formula: see text], the isomorphism problem is solved: we describe multiplications from [Formula: see text] that define isomorphic rings on [Formula: see text]. We also describe Abelian groups that are realized as the multiplication group of some group in [Formula: see text]. We describe groups in [Formula: see text] that are isomorphic to their multiplication groups.