Multiplications on torsion-free groups of finite rank

E. I. Kompantseva ; A. Tuganbaev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 219 (2023), pp. 3-15

Résumé

A multiplication on an Abelian group $G$ is an arbitrary homomorphism $\mu\colon G\otimes G\rightarrow G$. The set $\operatorname{Mult}G$ 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 $\mathcal{A}_0$ 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 $\mathcal{A}_0$, the group $\operatorname{Mult}G$ also belongs to this class. The rank, regulator, regulator index, almost isomorphism invariants, principal decomposition, and standard representation of the group $\operatorname{Mult}G$ for $G\in \mathcal{A}_0$ are described.

Détail

Keywords: Abelian group, almost completely decomposable Abelian group, ring on an Abelian group
Mots-clés : multiplication group of an Abelian group.