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Absolute Nil-Ideals of Abelian Groups

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.

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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). https://doi.org/10.1007/s10958-014-1745-y

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Keywords

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