ON <i>SI</i>-GROUPS

Andruszkiewicz, Ryszard; Woronowicz, Mateusz

doi:10.1017/s0004972714000641

Cited by 7 publications

(10 citation statements)

References 9 publications

(22 reference statements)

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“…Such groups are called SI-groups. In [5] we have noted and corrected some inconsistencies in Feigelstock's paper and we have presented new results concerning the structure of SI-groups. The aim of the first part of that note is to continue our studies with stronger assumptions.…”

Section: Introductionmentioning

confidence: 90%

“…Remark 3.2. In [5] we have introduced a new necessary terminology to describe SI-groups. We remind the reader that an abelian group A is called an SI H -group, if every associative ring R with R + = A is an H-ring (i.e., an associative ring in which all subrings are two-sided ideals).…”

Section: Sgi -Groupsmentioning

confidence: 99%

“…Hence the property S, in conjunction with property SI H gives precisely the property SGI. For preliminary knowledge on SI H -groups we refer the reader to [5,8].…”

Section: Sgi -Groupsmentioning

confidence: 99%

“…It follows from Remark 3.2 and Theorem 2.5 that an abelian torsion-free group is an SGI-group if and only if it is an S-group. The classification of torsion-free SGI-groups follows also from[5, Theorem 3.10]. Theorem 2.10 and Remark 3.2 imply at once that there does not exist any mixed SGI-group.…”

mentioning

confidence: 94%

“…The next two section follow partially from results achieved in [5,8] and they are related to the notion of SR-group introduced in [2].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Note on Additive Groups of Some Specific Associative Rings

Woronowicz

2016

Annales Mathematicae Silesianae

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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.

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Section: Introductionmentioning

confidence: 90%

Section: Sgi -Groupsmentioning

confidence: 99%

“…Hence the property S, in conjunction with property SI H gives precisely the property SGI. For preliminary knowledge on SI H -groups we refer the reader to [5,8].…”

Section: Sgi -Groupsmentioning

confidence: 99%

mentioning

confidence: 94%

“…The next two section follow partially from results achieved in [5,8] and they are related to the notion of SR-group introduced in [2].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Note on Additive Groups of Some Specific Associative Rings

Woronowicz

2016

Annales Mathematicae Silesianae

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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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On the Square Subgroup of a MixedSI-Group

Andruszkiewicz¹,

Woronowicz²

2018

Proceedings of the Edinburgh Mathematical Society

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The relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

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ON SI-GROUPS

Cited by 7 publications

References 9 publications

A Note on Additive Groups of Some Specific Associative Rings

A Note on Additive Groups of Some Specific Associative Rings

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

On the Square Subgroup of a MixedSI-Group

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