Square Submodule of a Module

Annales Mathematicae Silesianae

2016

Almost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

“…If we restrict our consideration to associative rings R with R + = A, then we write a A. More information about square subgroups is available in [1,3].…”

Section: Introductionmentioning

confidence: 99%

A Note on Additive Groups of Some Specific Associative Rings

Annales Mathematicae Silesianae

2016

“…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%

“…The notion was partially investigated by Aghdam in [1] and it is closely connected with the paper [16] by Stratton and Webb. Aghdam continued his research on the square subgroup together with Najafizadeh in [2][3][4]. Nevertheless, the basic question related to the topic remained unanswered.…”

Section: Introductionmentioning

confidence: 99%

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

Andruszkiewicz

2016

Bull. Aust. Math. Soc.

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.

“…However, the replacement of the subgroup H by A in Feigelstock's problem made it much harder. The problem was unsolved for 35 years although it appeared in papers related to this issue (see, [1,[3][4][5]). Negative answers for mixed and torsion-free abelian groups were given by A. Najafizadeh, R.R.…”

Section: Introductionmentioning

confidence: 99%

“…Previously, the square subgroup of a torsion-free abelian group was investigated only in some special cases. Namely, A.M. Aghdam and A. 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]). In particular, they have described the square subgroups of these groups.…”

Section: Introductionmentioning

confidence: 99%

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

Annales Mathematicae Silesianae

2018

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.