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$\tau $-supplemented modules and $\tau $-weakly supplemented modules

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*
T. M. Kosan
*

** Address.**
Faculty of Science, Gebze Institute of Techonology, Cayirova Campus, 41400 Gebze- Kocaeli, Turkiye

** E-mail:**
mtkosan@gyte.edu.tr

**Abstract.**
Given a hereditary torsion theory $\tau
= (\mathbb{T},\mathbb{F})$ in Mod-$R$, a module $M$ is called {\it
$\tau$-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 {\it $\tau$-supplement\/} of $U$ in $M$ if
$U+V=M$ and $U\cap V\leq \tau (V)$ and $M$ is {\it $\tau$-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 $\tau (M_2)\leq_e M_2$. Also, it
is shown that; any finite sum of $\tau$-weakly supplemented
modules is a~$\tau$-weakly supplemented module.

**AMSclassification. ** 16D50, 16L60.

**Keywords. ** Torsion theory, $\tau$-supplement submodule.