On the Square Subgroup of a Mixed SI-Group
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...
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| Published in | Proceedings of the Edinburgh Mathematical Society Vol. 61; no. 1; pp. 295 - 304 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.02.2018
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| Online Access | Get full text |
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| Abstract | 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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| AbstractList | 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. Abstract 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. |
| Author | Woronowicz, M. Andruszkiewicz, R. R. |
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| Cites_doi | 10.1080/00927872.2015.1044107 10.1007/978-94-017-0339-0 10.1017/S0004972714000641 10.4153/CJM-1968-083-1 10.1007/s00009-010-0041-4 10.1017/S1446788713000268 10.1007/BF01111051 10.1081/AGB-120018987 10.1017/S0004972700034110 10.1007/BF02027824 10.5486/PMD.1956.4.3-4.50 10.5486/PMD.1980.27.1-2.16 10.4064/cm117-1-2 |
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| Keywords | 17D99 SI-groups Primary 20K99 the square subgroup of an abelian group non-splitting abelian groups Secondary 16D25 additive groups of rings |
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| References | Stratton (S0013091517000165_ref20) 1980; 27 Fuchs (S0013091517000165_ref14) 1970; 1 S0013091517000165_ref3 S0013091517000165_ref2 S0013091517000165_ref5 S0013091517000165_ref17 S0013091517000165_ref4 S0013091517000165_ref7 S0013091517000165_ref16 S0013091517000165_ref6 Najafizadeh (S0013091517000165_ref18) 2015; 27 S0013091517000165_ref9 Feigelstock (S0013091517000165_ref11) 1983; 1 Feigelstock (S0013091517000165_ref12) 1988; 2 Fuchs (S0013091517000165_ref13) 1956; 4 S0013091517000165_ref19 S0013091517000165_ref10 Aghdam (S0013091517000165_ref1) 1987; 51 Balcerzyk (S0013091517000165_ref8) 1989 Fuchs (S0013091517000165_ref15) 1973; 2 |
| References_xml | – ident: S0013091517000165_ref7 doi: 10.1080/00927872.2015.1044107 – volume: 2 volume-title: Infinite Abelian groups year: 1973 ident: S0013091517000165_ref15 contributor: fullname: Fuchs – ident: S0013091517000165_ref9 doi: 10.1007/978-94-017-0339-0 – volume: 51 start-page: 343 year: 1987 ident: S0013091517000165_ref1 article-title: Square subgroup of an Abelian group publication-title: Acta. Sci. Math. contributor: fullname: Aghdam – volume: 2 volume-title: Additive groups of rings year: 1988 ident: S0013091517000165_ref12 contributor: fullname: Feigelstock – ident: S0013091517000165_ref6 doi: 10.1017/S0004972714000641 – ident: S0013091517000165_ref17 doi: 10.4153/CJM-1968-083-1 – volume: 1 volume-title: Additive groups of rings year: 1983 ident: S0013091517000165_ref11 contributor: fullname: Feigelstock – ident: S0013091517000165_ref3 doi: 10.1007/s00009-010-0041-4 – ident: S0013091517000165_ref5 doi: 10.1017/S1446788713000268 – ident: S0013091517000165_ref16 doi: 10.1007/BF01111051 – ident: S0013091517000165_ref4 doi: 10.1081/AGB-120018987 – volume: 27 start-page: 1 year: 2015 ident: S0013091517000165_ref18 article-title: On the square submodule of a mixed module publication-title: Gen. Math. Notes contributor: fullname: Najafizadeh – ident: S0013091517000165_ref10 doi: 10.1017/S0004972700034110 – ident: S0013091517000165_ref19 doi: 10.1007/BF02027824 – volume: 1 volume-title: Infinite Abelian groups year: 1970 ident: S0013091517000165_ref14 contributor: fullname: Fuchs – volume: 4 start-page: 488 year: 1956 ident: S0013091517000165_ref13 publication-title: Publ. Math. Debrecen doi: 10.5486/PMD.1956.4.3-4.50 contributor: fullname: Fuchs – volume-title: Commutative Noetherian and Krull rings year: 1989 ident: S0013091517000165_ref8 contributor: fullname: Balcerzyk – volume: 27 start-page: 127 year: 1980 ident: S0013091517000165_ref20 article-title: Abelian groups, nil modulo a subgroup, need not have nil quotient group publication-title: Publ. Math. Debrecen. doi: 10.5486/PMD.1980.27.1-2.16 contributor: fullname: Stratton – ident: S0013091517000165_ref2 doi: 10.4064/cm117-1-2 |
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| Title | On the Square Subgroup of a Mixed SI-Group |
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