Geodesic
Absolute nil-ideals of Abelian groups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 8, pp. 63-76 
E. I. Kompantseva. Absolute nil-ideals of Abelian groups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 8, pp. 63-76. http://geodesic.mathdoc.fr/item/FPM_2012_17_8_a8/
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     author = {E. I. Kompantseva},
     title = {Absolute nil-ideals of {Abelian} groups},
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is known that in an Abelian group $G$ that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on $G$ is $\bigcap_ppT(G)$, where $T(G)$ is the torsion part of $G$. In this work, we define a pure fully invariant subgroup $G^*\supseteq T(G)$ of an arbitrary Abelian mixed group $G$ and prove that if $G$ contains no nonzero torsion-free subgroups, then the subgroup $\bigcap_ppG^*$ is a nil-ideal in any ring on $G$, and the first Ulm subgroup $G^1$ is its nilpotent ideal.

[1] Dzhekobson N., Stroenie kolets, IL, M., 1961

[2] Kompantseva E. I., “Koltsa bez krucheniya”, Fundament. i prikl. mat., 15:8 (2009), 95–143 | MR

[3] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir, M., 1974; т. 2, 1977

[4] Fried E., “On the subgroups of Abelian groups that are ideals in every ring”, Proc. Colloq. Abelian Groups, Budapest, 1964, 51–55 | MR | Zbl

[5] Gardner B. J., “Rings on completely decomposable torsion-free Abelian groups”, Comment. Math. Univ. Carolin., 15:3 (1974), 381–382 | MR

[6] Gardner B. J., Jackett D. R., “Rings on certain classes of torsion-free Abelian groups”, Comment. Math. Univ. Carolin., 17:3 (1976), 439–506 | MR

[7] Jackett D. R., “Rings on certain mixed Abelian groups”, Pacific J. Math., 98:2 (1982), 355–373 | DOI | MR

[8] Topics in Abelian Groups, Chicago, 1963

[9] Toubassi E. H., Lawver D. A., “Height-slope and splitting length of Abelian groups”, Publ. Mat., 20 (1973), 63–71 | MR | Zbl