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For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrowG$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of allmultiplications on an Abelian group $G$ is an Abelian group with respect toaddition. An Abelian group $G$ with multiplication, defined on it, is called a\textsf{ring on the group} $G$. Let $\mathcal{A}_0$ be the class of Abelianblock-rigid almost completely decomposable groups of ring type with cyclicregulator quotient. In the paper, we study relationships between the abovegroups and their multiplication groups. It is proved that groups from$\mathcal{A}_0$ are definable by their multiplication groups. For a rigid group$G\in\mathcal{A}_0$, the isomorphism problem is solved: we describemultiplications from $\text{Mult}\,G$ that define isomorphic rings on $G$. Wedescribe Abelian groups that are realized as the multiplication group of somegroup in $\mathcal{A}_0$. We also describe groups in $\mathcal{A}_0$ that areisomorphic to their multiplication groups.

Relationships between Almost Completely Decomposable Abelian Groups with Their Multiplication Groups