Vol 219 (2023)

A multiplication on an Abelian group G is an arbitrary homomorphism μ: G ⊗ G → G. The set MultG of all multiplications on an Abelian group G is itself an Abelian group with respect to addition. In this paper, we discuss the multiplication groups of groups from the class A₀ of all Abelian block-rigid, almost completely decomposable groups of ring type with cyclic regulatory factors. We show that for any group G from the class A₀, the group MultG also belongs to this class. The rank, regulator, regulator index, almost isomorphism invariants, principal decomposition, and standard representation of the group MultG for G ∈ A₀ are described.

We consider realization and isomorphism problems for formal matrix rings over a given ring. Principal multiplier matrices of such rings play an important role in this case.

We show that a ring R with center Z(R) such that the module RZ(R) is an essential extension of the module Z(R)Z(R) need not be right quasi-invariant, i.e., not all maximal right ideals of the ring R are ideals. In terms of the central essentiality property, we obtain sufficient conditions for the fact that all maximal right ideals are ideals.

In this survey, we systematically examine rings and semirings that are either commutative or satisfy the following condition: for any noncentral element a, there exist nonzero central elements x and y such that ax = y.