A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

Mateusz Woronowicz

doi:10.1515/amsil-2017-0009

Abstract

AbstractA hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

Highlights

The most natural issue concerning the abelian groups in the context of defining the ring structure on them, is the following question: given an abelian group A, does there exist a ring (A, ∗) satisfying A∗A = {0}? If the answer is negative, A is called a nil-group
Given an abelian group A, the square subgroup A of A can be understood as the subgroup of A generated by squares of all possible rings defined on A
The authors have studied there the square subgroup of a torsion-free abelian group A = A1 ⊕ A2 of rank three, assuming that Ai is a group of rank i, A2 is not a nil group and either t(A1) ∈ T (A2) or t(A1) is incomparable to any type belonging to T (A2)

Summary

Introduction

Najafizadeh have proved that for every indecomposable torsion-free abelian group A of rank two which is not homogeneous, the quotient group A/ A is a nil group (see, [4,5]) They have described the square subgroups of these groups. The authors have studied there the square subgroup of a torsion-free abelian group A = A1 ⊕ A2 of rank three, assuming that Ai is a group of rank i, A2 is not a nil group and either t(A1) ∈ T (A2) or t(A1) is incomparable to any type belonging to T (A2) They have achieved interesting results in this field which, in some cases, contribute to this area of research. We give a description of the square subgroup of a completely decomposable torsion-free abelian group (in both cases of associative and general rings). We make a little correction of [16, Theorem 3.7] (see, Remark 4.10) and we prove that aA is a characteristic subgroup of A

Notation and preliminaries

Findings

Other results