| Title:
|
$\tau $-supplemented modules and $\tau $-weakly supplemented modules (English) |
| Author:
|
Koşan, Muhammet Tamer |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
43 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
251-257 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
torsion theory |
| Keyword:
|
$\tau $-supplement submodule |
| MSC:
|
16D10 |
| MSC:
|
16D50 |
| MSC:
|
16L60 |
| idZBL:
|
Zbl 1156.16006 |
| idMR:
|
MR2378525 |
| . |
| Date available:
|
2008-06-06T22:51:29Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108069 |
| . |
| Reference:
|
[1] Anderson F. W., Fuller K. R.: Rings and Categories of Modules.Springer-Verlag, New York, 1992. Zbl 0765.16001, MR 1245487 |
| Reference:
|
[2] Clark J., Lomp C., Vanaja N., Wisbauer R.: Lifting Modules.Birkhäuser, Basel, 2006. Zbl 1102.16001, MR 2253001 |
| Reference:
|
[3] Golan J. S.: Torsion Theories.Pitman Monographs and Surveys in Pure and Applied Mathematics 29, New York, John Wiley & Sons, 1986. Zbl 0657.16017, MR 0880019 |
| Reference:
|
[4] Koşan T., Harmanci A.: Modules supplemented with respect to a torsion theory.Turkish J. Math. 28 (2), (2004), 177–184. MR 2062562 |
| Reference:
|
[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 2115677 |
| Reference:
|
[6] Koşan M. T., Harmanci A.: $\oplus $-supplemented modules relative to a torsion theory.New-Zealand J. Math. 35 (2006), 63–75. Zbl 1104.16026, MR 2222176 |
| Reference:
|
[7] Mohamed S. H., Müller B. J.: Continuous and discrete modules.London Math. Soc. LNS 147, Cambridge Univ. Press, Cambridge (1990). Zbl 0701.16001, MR 1084376 |
| Reference:
|
[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 1437673 |
| Reference:
|
[9] Stenström B.: Rings of quotients.Springer Verlag, Berlin, 1975. MR 0389953 |
| Reference:
|
[10] Wisbauer R.: Foundations of module and ring theory.Gordon and Breach, Reading, 1991. Zbl 0746.16001, MR 1144522 |
| Reference:
|
[11] Zhou Y.: Generalizations of perfect, semiperfect, and semiregular rings.Algebra Colloquium 7 (3), (2000), 305–318. Zbl 0994.16016, MR 1810586 |
| . |
