arXiv subjects

Ekaterina Kompantseva

Publications and source records attributed to Ekaterina Kompantseva.

4 recordsLinked to original sources

arXiv2023

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.

math.GR↗

arXiv2023

Relationships between Almost Completely Decomposable Abelian Groups with Their Multiplication Groups

For an Abelian group $G$, any homomorphism $μ\colon G\otimes G\rightarrow G$ 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.

math.GR↗

arXiv2022

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

For an Abelian group $G$, any homomorphism $μ\colon G\otimes G\rightarrow G$ 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 $\text{Mult}\,G$.

math.GR↗

arXiv2021

Rings on Abelian Torsion-Free Groups of Finite Rank

In the class of reduced Abelian torsion-free groups $G$ of finite rank, we describe TI-groups, this means that every associative ring on $G$ is filial. If every associative multiplication on $G$ is the zero multiplication, then $G$ is called a $nil_a$-group. It is proved that a reduced Abelian torsion-free group $G$ of finite rank is a $TI$-group if and only if $G$ is a homogeneous Murley group or $G$ is a $nil_a$-group. We also study the interrelations between the class of homogeneous Murley groups and the class of $nil_a$-groups. For any type $t\ne (\infty,\infty,\ldots)$ and every integer $n>1$, there exist $2^{\aleph_0}$ pairwise non-quasi-isomorphic homogeneous Murley groups of type $t$ and rank $n$ which are $nil_a$-groups. We describe types $t$ such that there exists a homogeneous Murley group of type $t$ which is not a $nil_a$-group. This paper will be published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.

math.RA↗