On quasi-endomorphism rings of some strongly indecomposable torsion-free Abelian groups of rank~4

A. V. Cherednikova

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 245-255

Résumé

By the quasi-endomorphism ring $ \mathcal{E}(G)$ of a torsion-free Abelian group $G$ of finite rank we mean divisible hull of the endomorphism ring of the group. The elements of $\mathcal{E}(G)$ is called quasi-endomorphisms of $G$. Thus the quasi-endomorphisms of the group $G$ is normal endomorphisms, which formally divided by non-zero integers. In the paper it is considered quasi-endomorphism rings of class of strongly indecomposable torsion-free Abelian groups of rank 4 with one $\tau$-adic relation, whose pseudo-socles have rank 1. Let $\tau = [(m_p)]$ be a fixed type, where $m_p$ is a non-negative integer or the symbol $\infty $, indexed by elemets of $P$, the set of primes numbers. Denote by $K_p = \mathbb{Z} _ {p ^ {m_p }}$ the residue class ring modulo $p^{m_p}$ in the case $m_p \infty$ and ring of $p$-adic integers if $m_p = \infty$. We use the description of the groups from the above class up to quasi-isomorphism in terms of four-dimension over the field of rational numbers $\mathbb{Q}$ subspaces of algebra $\mathbb{Q}(\tau) = \mathbb{Q} \otimes \prod_ {p\,\in p} K_{p}$. The existing relationship between the quasi-endomorphisms of a group $G$ of this class and endomorphisms of the corresponding of this group subspace $U$ of the algebra $\mathbb{Q}(\tau)$ allows us to represent the quasi-endomorphisms of the group $G$ in the form of a matrices of order 4 over the field of rational numbers. In this paper, a classification of the quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with one $\tau$-adic relation, whose pseudo-socles have rank 1, is obtained. It is proved that, up to isomorphism, there exist two algebras and one infinite series of algebras with rational parameter, which are realized as quasi-endomorphism rings of groups of this class. Bibliography: 17 titles.

Détail

Keywords: quasi-endomorphism ring, Abelian group, torsion-free group of finite rank, strongly indecomposable group.