A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings

It is studied how rank two pure subgroups of a torsion-free Abelian group of rank three influences its structure and type set. In particular, the criterion for such a subgroup B to be a direct summand of a torsion-free Abelian group of rank three with the finite type set containing the greatest elem...

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Main Authors:	Najafizadeh Alireza, Woronowicz Mateusz
Format:	Article
Language:	English
Published:	Sciendo 2017-12-01
Series:	Discussiones Mathematicae - General Algebra and Applications
Subjects:	rank type square subgroup additive groups of rings 20k15 20k21 20k27
Online Access:	https://doi.org/10.7151/dmgaa.1274

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spelling	doaj-b0903dae1ded4cec9b1b141d18c663292021-09-05T17:19:43ZengSciendoDiscussiones Mathematicae - General Algebra and Applications2084-03732017-12-0137222323210.7151/dmgaa.1274dmgaa.1274A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative RingsNajafizadeh Alireza0Woronowicz Mateusz1Department of mathematics Payame Noor University, Teheran, IranInstitute of Mathematics University of Białystok, Białystok, PolandIt is studied how rank two pure subgroups of a torsion-free Abelian group of rank three influences its structure and type set. In particular, the criterion for such a subgroup B to be a direct summand of a torsion-free Abelian group of rank three with the finite type set containing the greatest element which does not belong to the type set of B, is presented. Some results for nil groups and the square subgroup of a decomposable torsion-free Abelian group are also achieved. Moreover, new results for mixed Abelian groups supporting only associative rings are obtained. In particular, the first example of an Abelian group supporting only associative rings but not only commutative rings is given.https://doi.org/10.7151/dmgaa.1274ranktypesquare subgroupadditive groups of rings20k1520k2120k27
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Najafizadeh Alireza Woronowicz Mateusz
spellingShingle	Najafizadeh Alireza Woronowicz Mateusz A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings Discussiones Mathematicae - General Algebra and Applications rank type square subgroup additive groups of rings 20k15 20k21 20k27
author_facet	Najafizadeh Alireza Woronowicz Mateusz
author_sort	Najafizadeh Alireza
title	A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
title_short	A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
title_full	A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
title_fullStr	A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
title_full_unstemmed	A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
title_sort	note on additive groups of some specific torsion-free rings of rank three and mixed associative rings
publisher	Sciendo
series	Discussiones Mathematicae - General Algebra and Applications
issn	2084-0373
publishDate	2017-12-01
description	It is studied how rank two pure subgroups of a torsion-free Abelian group of rank three influences its structure and type set. In particular, the criterion for such a subgroup B to be a direct summand of a torsion-free Abelian group of rank three with the finite type set containing the greatest element which does not belong to the type set of B, is presented. Some results for nil groups and the square subgroup of a decomposable torsion-free Abelian group are also achieved. Moreover, new results for mixed Abelian groups supporting only associative rings are obtained. In particular, the first example of an Abelian group supporting only associative rings but not only commutative rings is given.
topic	rank type square subgroup additive groups of rings 20k15 20k21 20k27
url	https://doi.org/10.7151/dmgaa.1274
work_keys_str_mv	AT najafizadehalireza anoteonadditivegroupsofsomespecifictorsionfreeringsofrankthreeandmixedassociativerings AT woronowiczmateusz anoteonadditivegroupsofsomespecifictorsionfreeringsofrankthreeandmixedassociativerings AT najafizadehalireza noteonadditivegroupsofsomespecifictorsionfreeringsofrankthreeandmixedassociativerings AT woronowiczmateusz noteonadditivegroupsofsomespecifictorsionfreeringsofrankthreeandmixedassociativerings
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A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings

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