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Absolute Nil-Ideals of Abelian Groups


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.

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  1. 1.

    E. Fried, “On the subgroups of Abelian groups that are ideals in every ring,” in: Proc. Colloq. Abelian Groups, Budapest (1964), pp. 51–55.

  2. 2.

    L. Fuchs, Infinite Abelian Groups, Vol. 2, Academic Press, New York (1977).

    Google Scholar 

  3. 3.

    B. J. Gardner, “Rings on completely decomposable torsion-free Abelian groups,” Comment. Math. Univ. Carolin., 15, No. 3, 381–382 (1974).

    MATH  Google Scholar 

  4. 4.

    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).

    MathSciNet  Google Scholar 

  5. 5.

    D. R. Jackett, “Rings on certain mixed Abelian groups,” Pacific J. Math., 98, No. 2, 355–373 (1982).

    Article  MathSciNet  Google Scholar 

  6. 6.

    N. Jacobson, Structure of Rings [Russian translation], Inostr. Lit., Moscow (1961).

    Google Scholar 

  7. 7.

    E. I. Kompantseva, “Torsion-free rings,” J. Math. Sci., 171, No. 2, 213–247 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Topics in Abelian Groups, Chicago (1963).

  9. 9.

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

    MATH  MathSciNet  Google Scholar 

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Correspondence to E. I. Kompantseva.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 63–76, 2011/12.

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Kompantseva, E.I. Absolute Nil-Ideals of Abelian Groups. J Math Sci 197, 625–634 (2014).

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  • Abelian Group
  • Quotient Group
  • Torsion Group
  • Invariant Subgroup
  • Pure Subgroup