Geodesic
Quotient Divisible Groups of Rank~2
Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 37-51.

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

In the paper, representations of torsion-free Abelian groups of rank $2$ using torsion-free groups of rank $1$ are studied. Necessary and sufficient conditions are found under which a group given by such a representation is quotient divisible. A criterion is obtained for two $p$-minimal quotient divisible torsion-free groups of rank $2$ to be isomorphic to each other. An example is constructed showing that two such groups can be embedded in each other but be not isomorphic. A series of properties of fundamental systems of elements of quotient divisible groups of arbitrary finite rank is established.
Keywords: Abelian group, group of rank $2$.
Mots-clés : quotient divisible group, quotient divisible envelope 
@article{MZM_2021_110_1_a3,
     author = {M. N. Zonov and E. A. Timoshenko},
     title = {Quotient {Divisible} {Groups} of {Rank~2}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {37--51},
     publisher = {mathdoc},
     volume = {110},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a3/}
}
TY  - JOUR
AU  - M. N. Zonov
AU  - E. A. Timoshenko
TI  - Quotient Divisible Groups of Rank~2
JO  - Matematičeskie zametki
PY  - 2021
SP  - 37
EP  - 51
VL  - 110
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a3/
LA  - ru
ID  - MZM_2021_110_1_a3
ER  - 
%0 Journal Article
%A M. N. Zonov
%A E. A. Timoshenko
%T Quotient Divisible Groups of Rank~2
%J Matematičeskie zametki
%D 2021
%P 37-51
%V 110
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a3/
%G ru
%F MZM_2021_110_1_a3
M. N. Zonov; E. A. Timoshenko. Quotient Divisible Groups of Rank~2. Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 37-51. http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a3/

[1] L. Fuchs, Abelian Groups, Springer, Cham, 2015 | MR

[2] R. A. Beaumont, R. S. Pierce, “Torsion-free rings”, Illinois J. Math., 5:1 (1961), 61–98 | DOI | MR

[3] A. Fomin, W. Wickless, “Quotient divisible abelian groups”, Proc. Amer. Math. Soc., 126:1 (1998), 45–52 | DOI | MR

[4] A. A. Fomin, “Abelevy gruppy so svobodnymi podgruppami beskonechnogo indeksa i ikh koltsa endomorfizmov”, Matem. zametki, 36:2 (1984), 179–187 | MR | Zbl

[5] A. V. Tsarev, “Psevdoratsionalnyi rang faktorno delimoi gruppy”, Fundament. i prikl. matem., 11:3 (2005), 201–213 | MR | Zbl

[6] O. I. Davydova, “Faktorno delimye gruppy ranga 1”, Fundament. i prikl. matem., 13:3 (2007), 25–33 | MR | Zbl

[7] A. Fomin, “Invariants for abelian groups and dual exact sequences”, J. Algebra, 322:7 (2009), 2544–2565 | DOI | MR

[8] A. V. Tsarev, “Modul psevdoratsionalnykh otnoshenii faktorno delimoi gruppy”, Algebra i analiz, 22:1 (2010), 223–239 | MR | Zbl

[9] O. S. Guseva, A. V. Tsarev, “Koltsa, $p$-rangi kotorykh ne prevoskhodyat 1”, Matem. sb., 205:4 (2014), 21–32 | DOI | MR | Zbl

[10] A. A. Fomin, “K teorii faktorno delimykh grupp. I”, Fundament. i prikl. matem., 17:8 (2012), 153–167

[11] A. A. Fomin, “K teorii faktorno delimykh grupp. II”, Fundament. i prikl. matem., 20:5 (2015), 157–196 | MR

[12] L. Fuchs, “On subdirect unions, I”, Acta Math. Acad. Sci. Hungar., 3:1-2 (1952), 103–120 | DOI | MR

[13] R. A. Beaumont, R. J. Wisner, “Rings with additive group which is a torsion-free group of rank two”, Acta Sci. Math. (Szeged), 20:2-3 (1959), 105–116 | MR

[14] R. A. Beaumont, R. S. Pierce, “Torsion free groups of rank two”, Mem. Amer. Math. Soc., 38 (1961), Amer. Math. Soc., Providence, RI | MR

[15] D. M. Arnold, Finite Rank Torsion Free Abelian Groups and Rings, Lecture Notes in Math., 931, Springer, Berlin, 1982 | DOI | MR