Relationships between Almost Completely Decomposable Abelian Groups with Their Multiplication Groups

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.Comment: arXiv admin note: text overlap with arXiv:2205.1065

Absolute Ideals of Almost Completely Decomposable Abelian Groups

We consider the class $\mathcal{A}_0$ of Abelian block-rigid $CRQ$-groups of ring type. A subgroup $A$ of an Abelian group $G$ is called an \textsf{absolute ideal} of the group $G$ if $A$ is an ideal in any ring on $G$. We describe principal absolute ideals of groups in $\mathcal{A}_0$. This allows to prove that any group in $\mathcal{A}_0$ is an $afi$-group, i.e., a group $G$ such that any absolute ideal of $G$ is a fully invariant subgroup

Умножения на смешанных абелевых группах

A multiplication on an abelian group G is a homomorphism $$\mu: G\otimes G\rightarrow G$$. An mixed abelian group G is called an MT-group if every multiplication on the torsion part of the group G can be extended  uniquely to a multiplication on G. MT-groups have been studied in many articles on the theory of additive groups of rings, but their complete description has not yet been obtained. In this paper, a pure fully invariant subgroup $$G^*_\Lambda$$ is considered for an abelian MT-group G. One of the main properties of this subgroup is that $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ is a nil-ideal in every ring with the additive group G (here $$\Lambda (G)$$ is the set of all primes p, for which the p-primary component of G is non-zero). It is shown that for every MT-group G either $$G=G^*_\Lambda$$ or the quotient group $$G/G^*_\Lambda$$ is uncountable.Умножение на абелевой группе G - это гомоморфизм $$\mu: G\otimes G\rightarrow G$$. Абелева группа G называется MT-группой, если любое умноженеие на ее периодической части однозначно продолжается до умножения на G. MT-группы изучались во многих работах по теории аддитивных групп колец, но вопрос об их строении остается открытым. В настоящней работе для MT-группы G рассматривается сервантная вполне характеристическая подгруппа $$G^*_\Lambda$$, одно из основных свойств которой заключается в том, что  подгруппа $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ является ниль-идеалом в любом кольце с аддитивной группой G (здесь $$\Lambda(G)$$ - множество всех простых чисел p, для которых p-примарная компонента группы G отлична от нуля). Показано, что для любой MT-группы G либо $$G=G^*_\Lambda$$, либо факторгруппа $$G/G^*_\Lambda$$ несчетна

Факторно делимые группы и группы без кручения, соответствующие конечным абелевым группам

.

Алгебраически компактные абелевы TI-группы

An abelian group G is called a TI-group if every associative ring with additive group G is filial. An abelian group G such that every (associative) ring with additive group G is an SI-ring (a hamiltonian ring) is called an SI-group (an $$SI_H$$-group). In this paper, TI-groups, as well as SI-groups and $$SI_H$$-groups are described in the class of reduced algebraically compact abelian groups.Абелева группа G называется TI-группой если любое ассоциативное кольцо с аддитивной группой G является филиальным. Абелева группа называется SI-группой ($$SI_H$$-группой), если любое (ассоциативное) кольцо с аддитивной группой G является SI-кольцом (гамильтоновым кольцом). В работе в классе редуцированных алгебраически компактных абелевых групп описаны  TI-группы, а также SI-группы и $$SI_H$$-группы

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

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 $G$ itself is an Abelian group with respect to addition; the group is called the \textsf{multiplication group} of $G$. Let $\mathcal{A}_0$ be the class of all reduced block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, for groups $G\in \mathcal{A}_0$, we describe groups $\text{Mult}\,G$. We prove that for $G\in \mathcal{A}_0$, the group $\text{Mult}\,G$ also belongs to the class $\mathcal{A}_0$. For any group $G\in \mathcal{A}_0$, we describe the rank, the regulator, the regulator index, invariants of near-isomorphism, a main decomposition, and a standard representation of the group $\text{Mult}\,G$