FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 8, PAGES 63-76

## Absolute nil-ideals of Abelian groups

E. I. Kompantseva

Abstract

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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 $\bigcap_p pT\left(G\right)$, where $T\left(G\right)$ is the torsion part of $G$. In this work, we define a pure fully invariant subgroup $G$* Ê T(G) of an arbitrary Abelian mixed group $G$ and prove that if $G$ contains no nonzero torsion-free subgroups, then the subgroup $\bigcap_p pG^*$ is a nil-ideal in any ring on $G$, and the first Ulm subgroup $G1$ is its nilpotent ideal.

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Last modified: March 23, 2013