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

Voir la notice de l'article

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/}
}
TY  - JOUR
AU  - Koşan, Muhammet Tamer
TI  - $\tau $-supplemented modules and $\tau $-weakly supplemented modules
JO  - Archivum mathematicum
PY  - 2007
SP  - 251
EP  - 257
VL  - 43
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a2/
LA  - en
ID  - ARM_2007_43_4_a2
ER  - 
%0 Journal Article
%A Koşan, Muhammet Tamer
%T $\tau $-supplemented modules and $\tau $-weakly supplemented modules
%J Archivum mathematicum
%D 2007
%P 251-257
%V 43
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2007_43_4_a2/
%G en
%F ARM_2007_43_4_a2
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/

[1] Anderson F. W., Fuller K. R.: Rings and Categories of Modules. Springer-Verlag, New York, 1992. | MR | Zbl

[2] Clark J., Lomp C., Vanaja N., Wisbauer R.: Lifting Modules. Birkhäuser, Basel, 2006. | MR | Zbl

[3] Golan J. S.: Torsion Theories. Pitman Monographs and Surveys in Pure and Applied Mathematics 29, New York, John Wiley & Sons, 1986. | MR | Zbl

[4] Koşan T., Harmanci A.: Modules supplemented with respect to a torsion theory. Turkish J. Math. 28 (2), (2004), 177–184. | MR

[5] Koşan M. T., Harmanci A.: Decompositions of Modules supplemented with respect to a torsion theory. Internat. J. Math. 16 (1), (2005), 43–52. | MR

[6] Koşan M. T., Harmanci A.: $\oplus $-supplemented modules relative to a torsion theory. New-Zealand J. Math. 35 (2006), 63–75. | MR | Zbl

[7] Mohamed S. H., Müller B. J.: Continuous and discrete modules. London Math. Soc. LNS 147, Cambridge Univ. Press, Cambridge (1990). | MR | Zbl

[8] Smith P. F., Viola-Prioli A. M., and Viola-Prioli J.: Modules complemented with respect to a torsion theory. Comm. Algebra 25 (1997), 1307–1326. | MR

[9] Stenström B.: Rings of quotients. Springer Verlag, Berlin, 1975. | MR

[10] Wisbauer R.: Foundations of module and ring theory. Gordon and Breach, Reading, 1991. | MR | Zbl

[11] Zhou Y.: Generalizations of perfect, semiperfect, and semiregular rings. Algebra Colloquium 7 (3), (2000), 305–318. | MR | Zbl