For an Abelian group G, any homomorphism ΞΌ:GβGβG is called a \textsf{multiplication} on G. The set MultG 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 A0β 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
A0β are definable by their multiplication groups. For a rigid group
GβA0β, the isomorphism problem is solved: we describe
multiplications from MultG that define isomorphic rings on G. We
describe Abelian groups that are realized as the multiplication group of some
group in A0β. We also describe groups in A0β that are
isomorphic to their multiplication groups.Comment: arXiv admin note: text overlap with arXiv:2205.1065