Geodesic
Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 60-76.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@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/}
}
TY  - JOUR
AU  - A. V. Cherednikova
TI  - Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2017
SP  - 60
EP  - 76
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2017_1_a6/
LA  - ru
ID  - IVM_2017_1_a6
ER  - 
%0 Journal Article
%A A. V. Cherednikova
%T Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2017
%P 60-76
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2017_1_a6/
%G ru
%F 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/

[1] Jonsson B., “On direct decompositions of torsion-free abelian groups”, Math. Scand., 7:2 (1959), 361–371 | DOI | MR

[2] Jonsson B., “On direct decompositions of torsion-free abelian groups”, Math. Scand., 5:2 (1957), 230–235 | DOI | MR | Zbl

[3] Reid J. D., “On the ring of quasi-endomorphisms of a torsion-free group”, Topics in Abelian groups, 1963, 51–68 | MR

[4] Baer R., “Abelian groups without elements of finite order”, Duke Math. J., 3:1 (1937), 68–122 | DOI | MR

[5] Beaumont R. A., Pierce R. S., “Torsion free groups of rank two”, Mem. Amer. Math. Soc., 38, 1961, 1–41 | MR

[6] Cherednikova A. V., “Koltsa kvaziendomorfizmov abelevykh pochti vpolne razlozhimykh grupp bez krucheniya ranga $3$”, Abelevy gruppy i moduli, 13–14 (1996), 237–242 | Zbl

[7] Cherednikova A. V., “Koltsa kvaziendomorfizmov kvazirazlozhimykh abelevykh grupp bez krucheniya ranga $3$”, Abelevy gruppy i moduli, 13–14 (1996), 224–236 | Zbl

[8] Cherednikova A. V., “Koltsa kvaziendomorfizmov silno nerazlozhimykh abelevykh grupp bez krucheniya ranga $3$”, Matem. zametki, 63:5 (1998), 763–773 | DOI | MR | Zbl

[9] Faticoni T. G., Direct sum decompositions of torsion-free finite rank groups, Chapman and Hall/CRC, Boca Raton, NY, 2007 | MR | Zbl

[10] Cherednikova A. V., “Koltsa kvaziendomorfizmov silno nerazlozhimykh abelevykh grupp bez krucheniya ranga $4$ s psevdotsokolyami ranga $3$”, Fundament. i prikl. matem., 16:3 (2010), 245–250 | MR | Zbl

[11] Cherednikova A. V., “O koltsakh kvaziendomorfizmov silno nerazlozhimykh abelevykh grupp bez krucheniya ranga $4$ s psevdotsokolyami ranga $1$”, Fundament. i prikl. matem., 17:8 (2012), 177–182 | Zbl

[12] Cherednikova A. V., “Koltsa kvaziendomorfizmov pochti vpolne razlozhimykh abelevykh grupp bez krucheniya ranga $4$ s nulevym radikalom Dzhekobsona”, Fundament. i prikl. matem., 17:8 (2012), 169–175

[13] Cherednikova A. V., “Koltsa kvaziendomorfizmov pochti vpolne razlozhimykh abelevykh grupp bez krucheniya ranga $4$, ne sovpadayuschikh so svoimi psevdotsokolyami”, Matem. zametki, 97:4 (2015), 609–619 | DOI | MR | Zbl

[14] Warfield R., “Homomorphisms and duality for torsion free groups”, Math. Z., 107:3 (1968), 189–200 | DOI | MR | Zbl

[15] Pirs R., Assotsiativnye algebry, Mir, M., 1986

[16] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir, M., 1974

[17] Fuks L., Beskonechnye abelevy gruppy, v. 2, Mir, M., 1977

[18] Fomin A. A., “Abelevy gruppy s odnim $\tau$-adicheskim sootnosheniem”, Algebra i logika, 28:1 (1989), 83–104 | MR | Zbl