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Title: Extending modules relative to a torsion theory (English)
Author: Doğruöz, Semra
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 58
Issue: 2
Year: 2008
Pages: 381-393
Summary lang: English
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Category: math
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Summary: An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau $-extending module, where $\tau $ is a hereditary torsion theory on $\mathop {\text{Mod}}$-$R$. An $R$-module $M$ is called type 2 $\tau $-extending if every type 2 $\tau $-closed submodule of $M$ is a direct summand of $M$. If $\tau _I$ is the torsion theory on $\mathop {\text{Mod}}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $\tau _I$-extending $R$-module, then the question of whether or not $M/MI$ is an extending $R/I$-module is investigated. In particular, for the Goldie torsion theory $\tau _G$ we give an example of a module that is type 2 ${\tau }_G$-extending but not extending. (English)
Keyword: torsion theory
Keyword: extending module
Keyword: closed submodule
MSC: 16D50
MSC: 16D70
MSC: 16D90
MSC: 16S90
idZBL: Zbl 1166.16014
idMR: MR2411096
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Date available: 2009-09-24T11:55:28Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/128264
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