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

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.