Geodesic
Extending modules relative to a torsion theory
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 381-393 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 16D50, 16D70, 16D90, 16S90
Keywords: torsion theory; extending module; closed submodule 
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     title = {Extending modules relative to a torsion theory},
     journal = {Czechoslovak Mathematical Journal},
     pages = {381--393},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a5/}
}
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Doğruöz, Semra. Extending modules relative to a torsion theory. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 381-393. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a5/

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