  For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow
G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all
multiplications on an Abelian group $G$ is an Abelian group with respect to
addition. 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 Abelian
block-rigid almost completely decomposable groups of ring type with cyclic
regulator quotient. In the paper, we study relationships between the above
groups 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 describe
multiplications from $\text{Mult}\,G$ that define isomorphic rings on $G$. We
describe Abelian groups that are realized as the multiplication group of some
group in $\mathcal{A}_0$. We also describe groups in $\mathcal{A}_0$ that are
isomorphic to their multiplication groups.
