Some New Results for the Square Subgroup of an Abelian Group

Andruszkiewicz, Ryszard; Woronowicz, Mateusz

doi:10.1080/00927872.2015.1044107

Cited by 10 publications

(8 citation statements)

References 3 publications

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“…Therefore he could not assume the associativity of rings, which is important for many algebraists. Our much more elementary proof in [6] allows the conclusion c 2016 Australian Mathematical Publishing Association Inc. 0004-9727/2016 $16.00 that Najafizadeh's result remains true also for the case of associative rings. It is a wellknown fact that there exists a torsion-free nil-group A such that A/nA is not a nilgroup for some positive integer n and, consequently, any ring R defined on A satisfies R 2 ⊆ nA (see [16]).…”

Section: Introductionmentioning

confidence: 68%

“…If we restrict our consideration to associative rings R with the additive group A, then the square subgroup of A is denoted by a A. It follows, from [6,Corollary 2.6], that if there exists an abelian group A which satisfies a A A, then A is reduced and nontorsion. More basic information about square subgroups and their generalisations is available in [1,3,6].…”

Section: Preliminariesmentioning

confidence: 99%

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A Torsion-Free Abelian Group Exists Whose Quotient Group Modulo the Square Subgroup Is Not A nil-Group

Andruszkiewicz

Woronowicz

2016

Bull. Aust. Math. Soc.

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The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the ring of $p$-adic integers are investigated using only elementary methods.

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Section: Introductionmentioning

confidence: 68%

Section: Preliminariesmentioning

confidence: 99%

A Torsion-Free Abelian Group Exists Whose Quotient Group Modulo the Square Subgroup Is Not A nil-Group

Andruszkiewicz

Woronowicz

2016

Bull. Aust. Math. Soc.

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“…where Mult (a) A means the set of all (associative) ring multiplications on the group A (compare with [6, Remarks 1.2 and 1.10]). It follows from [6,Corollary 2.6] that if there exists an abelian group A satisfying a A A, then A is reduced and non-torsion. For more detailed information on torsion-free groups of rank two and the square subgroup of various abelian groups we refer the reader to [1-3, 6, 7, 17].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…One of them is the concept of the square subgroup of an abelian 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 (see, [6]).…”

Section: Introductionmentioning

confidence: 99%

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

Woronowicz

2018

Annales Mathematicae Silesianae

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A 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.

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