Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 60-76
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We consider the a representation of quasi-endomorphisms of Abelian torsion-free groups of rank $4$ by matrices of order $4$ over the field of rational numbers $\mathbb{Q}$. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank $4$ quasi-decomposable into a direct sum of groups $A_1$, $A_2$ of rank $1$ and strongly indecomposable group $B$ of rank $2$ such that quasi-homomorphism groups $\mathbb {Q} \otimes \mathrm{Hom}(A_i, B)$ and $\mathbb {Q} \otimes \mathrm{Hom}(B, A_i)$ for any $i=1, 2$ have rank $1$ or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.
Keywords:
ring of quasi-endomorphisms, Abelian group
Mots-clés : torsion-free group, quasi-decomposable group.
Mots-clés : torsion-free group, quasi-decomposable group.
@article{IVM_2017_1_a6,
author = {A. V. Cherednikova},
title = {Rings of quasi-endomorphisms of some direct sums of torsion-free {Abelian} groups},
journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
pages = {60--76},
publisher = {mathdoc},
number = {1},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/IVM_2017_1_a6/}
}
A. V. Cherednikova. Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 60-76. http://geodesic.mathdoc.fr/item/IVM_2017_1_a6/