Geodesic
$\tau $-supplemented modules and $\tau $-weakly supplemented modules
Archivum mathematicum, Tome 43 (2007) no. 4, pp. 251-257 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Given a hereditary torsion theory $\tau = (\mathbb {T},\mathbb {F})$ in Mod-$R$, a module $M$ is called $\tau $-supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A/C$ $\tau -$torsion. A submodule $V$ of $M$ is called $\tau $-supplement of $U$ in $M$ if $U+V=M$ and $U\cap V\le \tau (V)$ and $M$ is $\tau $-weakly supplemented if every submodule of $M$ has a $\tau $-supplement in $M$. Let $M$ be a $\tau $-weakly supplemented module. Then $M$ has a decomposition $M=M_1\oplus M_2$ where $M_1$ is a semisimple module and $M_2$ is a module with $\tau (M_2)\le _e M_2$. Also, it is shown that; any finite sum of $\tau $-weakly supplemented modules is a $\tau $-weakly supplemented module.
Given a hereditary torsion theory $\tau = (\mathbb {T},\mathbb {F})$ in Mod-$R$, a module $M$ is called $\tau $-supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A/C$ $\tau -$torsion. A submodule $V$ of $M$ is called $\tau $-supplement of $U$ in $M$ if $U+V=M$ and $U\cap V\le \tau (V)$ and $M$ is $\tau $-weakly supplemented if every submodule of $M$ has a $\tau $-supplement in $M$. Let $M$ be a $\tau $-weakly supplemented module. Then $M$ has a decomposition $M=M_1\oplus M_2$ where $M_1$ is a semisimple module and $M_2$ is a module with $\tau (M_2)\le _e M_2$. Also, it is shown that; any finite sum of $\tau $-weakly supplemented modules is a $\tau $-weakly supplemented module.
Classification : 16D10, 16D50, 16L60
Keywords: torsion theory; $\tau $-supplement submodule 
@article{ARM_2007_43_4_a2,
     author = {Ko\c{s}an, Muhammet Tamer},
     title = {$\tau $-supplemented modules and $\tau $-weakly supplemented modules},
     journal = {Archivum mathematicum},
     pages = {251--257},
     year = {2007},
     volume = {43},
     number = {4},
     mrnumber = {2378525},
     zbl = {1156.16006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a2/}
}
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Koşan, Muhammet Tamer. $\tau $-supplemented modules and $\tau $-weakly supplemented modules. Archivum mathematicum, Tome 43 (2007) no. 4, pp. 251-257. http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a2/