A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

Abstract

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ℤ [p−1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.