| Title:
|
On $\mu $-singular and $\mu $-extending modules (English) |
| Author:
|
Talebi, Yahya |
| Author:
|
Hamzekolaee, Ali Reza Moniri |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
183-196 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $M$ be a module and $\mu $ be a class of modules in $\operatorname{Mod}-R$ which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a $\mu $-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu $. We define and investigate $\mu $-singular modules. We also introduce $\mu $-extending and weakly $\mu $-extending modules and mainly study weakly $\mu $-extending modules. We give some characterizations of $\mu $-co-H-rings by weakly $\mu $-extending modules. Let $R$ be a right non-$\mu $-singular ring such that all injective modules are non-$\mu $-singular, then $R$ is right $\mu $-co-H-ring if and only if $R$ is a QF-ring. (English) |
| Keyword:
|
$\mu $-essential submodule |
| Keyword:
|
$\mu $-singular module |
| Keyword:
|
$\mu $-extending module |
| Keyword:
|
weakly $\mu $-extending module |
| MSC:
|
16D10 |
| MSC:
|
16D70 |
| MSC:
|
16D99 |
| MSC:
|
16S90 |
| idMR:
|
MR2995871 |
| DOI:
|
10.5817/AM2012-3-183 |
| . |
| Date available:
|
2012-10-03T14:49:47Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142988 |
| . |
| Reference:
|
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| Reference:
|
[2] Dung, N. V., Huynh, D. V., Smith, P. F., Wisbauer, R.: Extending Modules.Pitman, London, 1994. |
| Reference:
|
[3] Faith, C.: Algebra II: Ring Theory.Springer–Verlag Berlin–Heidelberg–New York, 1976. MR 0427349 |
| Reference:
|
[4] Goodearl, K. R.: Ring Theory.Marcel Dekker, New York – Basel, 1976. MR 0429962 |
| Reference:
|
[5] Mohamed, S. H., Müller, B. J.: Continuous and Discrete Modules.London Math. Soc. 147 (1990). MR 1084376 |
| Reference:
|
[6] Oshiro, K.: Lifting modules, extending modules and their applications to QF-rings.Hokkaido Math. J. 13 (1984), 310–338. MR 0764267 |
| Reference:
|
[7] Özcan, A. Ç.: On GCO–modules and M–small modules.Comm. Fac. Sci. Univ. Ankara Ser. A1 51 (2) (2002), 25–36. Zbl 1038.16005, MR 1981050 |
| Reference:
|
[8] Özcan, A. Ç.: On $\mu $–essential and $\mu $–$M$–singular modules.Proceedings of the Fifth China–Japan–Korea Conference, Tokyo, Japan, 2007, pp. 272–283. MR 2513224 |
| Reference:
|
[9] Özcan, A. Ç.: The torsion theory cogenerated by $\delta $–M–small modules and GCO–modules.Comm. Algebra 35 (2007), 623–633. Zbl 1117.16020, MR 2294622, 10.1080/00927870601074871 |
| Reference:
|
[10] Talebi, Y., Vanaja, N.: The torsion theory cogenerated by M–small modules.Comm. Algebra 30 (3) (2002), 1449–1460. Zbl 1005.16029, MR 1892609, 10.1080/00927870209342390 |
| . |
