Geodesic
Sequences of Endomorphism Groups of Abelian Groups
Matematičeskie zametki, Tome 104 (2018) no. 2, pp. 309-317.

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

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite $p$-ranks. For an Abelian group $A$, a sequence of groups $(A_n)$ is considered, where $A_0=A$ and $A_{n+1}=\operatorname{End}A_n$. It is shown that, if all $p$-ranks of the group $A$ are finite, then this sequence can stabilize either after $A_0$ or after $A_1$.
Keywords: Abelian group, $E$-ring, $E$-group, $p$-rank. 
@article{MZM_2018_104_2_a13,
     author = {E. A. Timoshenko and A. V. Tsarev},
     title = {Sequences of {Endomorphism} {Groups} of {Abelian} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {309--317},
     publisher = {mathdoc},
     volume = {104},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a13/}
}
TY  - JOUR
AU  - E. A. Timoshenko
AU  - A. V. Tsarev
TI  - Sequences of Endomorphism Groups of Abelian Groups
JO  - Matematičeskie zametki
PY  - 2018
SP  - 309
EP  - 317
VL  - 104
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a13/
LA  - ru
ID  - MZM_2018_104_2_a13
ER  - 
%0 Journal Article
%A E. A. Timoshenko
%A A. V. Tsarev
%T Sequences of Endomorphism Groups of Abelian Groups
%J Matematičeskie zametki
%D 2018
%P 309-317
%V 104
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a13/
%G ru
%F MZM_2018_104_2_a13
E. A. Timoshenko; A. V. Tsarev. Sequences of Endomorphism Groups of Abelian Groups. Matematičeskie zametki, Tome 104 (2018) no. 2, pp. 309-317. http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a13/

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

[2] P. Schultz, “The endomorphism ring of the additive group of a ring”, J. Austral. Math. Soc., 15:1 (1973), 60–69 | DOI | MR | Zbl

[3] E. M. Kolenova, Opredelyaemost abelevykh grupp gruppami endomorfizmov, Dis. $\dots$ kand. fiz.-matem. nauk, Nizhnii Novgorod, 2006

[4] E. M. Kolenova, “Ob opredelyaemosti periodicheskoi $\mathrm{EndE}^+$-gruppy svoei gruppoi endomorfizmov”, Fundament. i prikl. matem., 13:2 (2007), 123–131 | MR | Zbl

[5] P. Grosse, “Periodizitat der Iterierten Homomorphismengruppen”, Arch. Math. (Basel), 16 (1965), 393–406 | DOI | MR | Zbl

[6] A. V. Grishin, A. V. Tsarëv, “$\mathcal E$-zamknutye gruppy i moduli”, Fundament. i prikl. matem., 17:2 (2012), 97–106 | MR | Zbl

[7] P. A. Krylov, A. V. Mikhalev, A. A. Tuganbaev, Endomorphism Rings of Abelian Groups, Algebr. Appl., 2, Kluwer Acad. Publ., Dordrecht, 2003 | MR | Zbl

[8] T. G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite Rank Groups, Pure Appl. Math. (Boca Raton), 285, CRC Press, Boca Raton, FL, 2007 | MR | Zbl

[9] C. Vinsonhaler, “$E$-rings and related structures”, Non-Noethereian Commutative Ring Theory, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2002, 387–402 | MR

[10] R. A. Bowshell, P. Schultz, “Unital rings whose additive endomorphisms commute”, Math. Ann., 228:3 (1977), 197–214 | DOI | MR | Zbl