Vol 219 (2023)
Статьи
Multiplications on torsion-free groups of finite rank
Abstract
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.
3-15
16-38
39-43
Centrally essential semirings
Abstract
A semiring is said to be centrally essential if, for every nonzero element x, there exist nonzero central elements y and z such that xy = z. We give several examples of noncommutative centrally essential semirings and describe some properties of additively cancellative, centrally essential semirings.
44-49
Maximal and minimal ideals of centrally essential rings
Abstract
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.
50-53
Centrally essential semigroup algebras
Abstract
For a cancellative semigroup S and a field F, we prove that the semigroup algebra FS is centrally essential if and only if the group of fractions GS of the semigroup S exists and the group algebra FGS of GS is centrally essential. The semigroup algebra of a cancellative semigroup is centrally essential if and only if it has the classical right ring of fractions, which is a centrally essential ring. There exist noncommutative, centrally essential semigroup algebras over fields of zero characteristic (this contrasts with the known fact that centrally essential group algebras over fields of zero characteristic are commutative).
54-59
Centrally essential rings and semirings
Abstract
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.
60-130