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.