τ -supplemented modules and τ -weakly supplemented modules

Muhammet Tamer Koşan

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 4, page 251-257
  • ISSN: 0044-8753

Abstract

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Given a hereditary torsion theory τ = ( 𝕋 , 𝔽 ) in Mod- R , a module M is called τ -supplemented if every submodule A of M contains a direct summand C of M with A / C τ - torsion. A submodule V of M is called τ -supplement of U in M if U + V = M and U ∩ V ≤ τ ( V ) and M is τ -weakly supplemented if every submodule of M has a τ -supplement in M . Let M be a τ -weakly supplemented module. Then M has a decomposition M = M 1 ⊕ M 2 where M 1 is a semisimple module and M 2 is a module with τ ( M 2 ) ≤ e M 2 . Also, it is shown that; any finite sum of τ -weakly supplemented modules is a  τ -weakly supplemented module.

How to cite

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Koşan, Muhammet Tamer. "$\tau $-supplemented modules and $\tau $-weakly supplemented modules." Archivum Mathematicum 043.4 (2007): 251-257. <http://eudml.org/doc/250161>.

@article{Koşan2007,
abstract = {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.},
author = {Koşan, Muhammet Tamer},
journal = {Archivum Mathematicum},
keywords = {torsion theory; $\tau $-supplement submodule; hereditary torsion theories; direct summands; weakly supplemented modules},
language = {eng},
number = {4},
pages = {251-257},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {$\tau $-supplemented modules and $\tau $-weakly supplemented modules},
url = {http://eudml.org/doc/250161},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Koşan, Muhammet Tamer
TI - $\tau $-supplemented modules and $\tau $-weakly supplemented modules
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 4
SP - 251
EP - 257
AB - 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.
LA - eng
KW - torsion theory; $\tau $-supplement submodule; hereditary torsion theories; direct summands; weakly supplemented modules
UR - http://eudml.org/doc/250161
ER -

References

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