Abstract
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 \(\bigcap\limits_{p} pT(G)\), where T(G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G* ⊇ T(G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup \(\bigcap\limits_{p} pG^{*}\) is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal.
Similar content being viewed by others
References
E. Fried, “On the subgroups of Abelian groups that are ideals in every ring,” in: Proc. Colloq. Abelian Groups, Budapest (1964), pp. 51–55.
L. Fuchs, Infinite Abelian Groups, Vol. 2, Academic Press, New York (1977).
B. J. Gardner, “Rings on completely decomposable torsion-free Abelian groups,” Comment. Math. Univ. Carolin., 15, No. 3, 381–382 (1974).
B. J. Gardner and D. R. Jackett, “Rings on certain classes of torsion free Abelian groups,” Comment. Math. Univ. Carolin., 17, No. 3, 439–506 (1976).
D. R. Jackett, “Rings on certain mixed Abelian groups,” Pacific J. Math., 98, No. 2, 355–373 (1982).
N. Jacobson, Structure of Rings [Russian translation], Inostr. Lit., Moscow (1961).
E. I. Kompantseva, “Torsion-free rings,” J. Math. Sci., 171, No. 2, 213–247 (2010).
Topics in Abelian Groups, Chicago (1963).
E. H. Toubassi and D. A. Lawver, “Height-slope and splitting length of Abelian groups,” Publ. Mat., 20, 63–71 (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 63–76, 2011/12.
Rights and permissions
About this article
Cite this article
Kompantseva, E.I. Absolute Nil-Ideals of Abelian Groups. J Math Sci 197, 625–634 (2014). https://doi.org/10.1007/s10958-014-1745-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1745-y