| Title:
|
On generalized q.f.d. modules (English) |
| Author:
|
Saleh, Mohammad |
| Author:
|
Jain, S. K. |
| Author:
|
López-Permouth, S. R. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
41 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
243-251 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A right $R$-module $M$ is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module $M$ is g.q.f.d. iff every direct sum of $M$-singular $M$-injective modules in ${\sigma [M]}$ is weakly injective iff every direct sum of $M$-singular weakly tight is weakly tight iff every direct sum of the injective hulls of $M$-singular simples is weakly $R$-tight. (English) |
| Keyword:
|
tight |
| Keyword:
|
weakly tight |
| Keyword:
|
weakly injective |
| Keyword:
|
q.f.d. |
| Keyword:
|
generalized q.f.d. modules |
| Keyword:
|
generalized weakly semisimple |
| MSC:
|
16D10 |
| MSC:
|
16D50 |
| MSC:
|
16D60 |
| MSC:
|
16D70 |
| MSC:
|
16D90 |
| idZBL:
|
Zbl 1114.16004 |
| idMR:
|
MR2188380 |
| . |
| Date available:
|
2008-06-06T22:46:05Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107955 |
| . |
| Reference:
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[1] Albu T., Nastasescu C.: Relative finiteness in module theory.Marcel Dekker 1984. Zbl 0556.16001, MR 0749933 |
| Reference:
|
[2] Al-Huzali A., Jain S. K., López-Permouth S. R.: Rings whose cyclics have finite Goldie dimension.J. Algebra 153 (1992), 37–40. MR 1195405 |
| Reference:
|
[3] Berry D.: Modules whose cyclic submodules have finite dimension.Canad. Math. Bull. 19 (1976), 1–6. Zbl 0335.16025, MR 0417244 |
| Reference:
|
[4] Brodskii G., Saleh M., Thuyet L., Wisbauer R.: On weak injectivity of direct sums of modules.Vietnam J. Math. 26 (1998), 121–127. MR 1684323 |
| Reference:
|
[5] Camillo V. P.: Modules whose quotients have finite Goldie dimension.Pacific J. Math. 69 (1977), 337–338. Zbl 0356.13003, MR 0442020 |
| Reference:
|
[6] Dhompong S., Sanwong J., Plubtieng S., Tansee H.: On modules whose singular subgenerated modules are weakly injective.Algebra Colloq. 8 (2001), 227–236. MR 1838519 |
| Reference:
|
[7] Dung N. V., Huynh D. V., Smith P., Wisbauer R.: Extending modules.Pitman, 1994. Zbl 0841.16001 |
| Reference:
|
[8] Golan J. S., López-Permouth S. R.: QI-filters and tight modules.Comm. Algebra 19 (1991), 2217–2229. MR 1123120 |
| Reference:
|
[9] Jain S. K., López-Permouth S. R.: Rings whose cyclics are essentially embeddable in projective modules.J. Algebra 128 (1990), 257–269. MR 1031920 |
| Reference:
|
[10] Jain S. K., López-Permouth S. R., Oshiro K., Saleh M.: Weakly projective and weakly injective modules.Canad. J. Math. 34 (1994), 972–981. MR 1295126 |
| Reference:
|
[11] Jain S. K., López-Permouth S. R., Singh S.: On a class of QI-rings.Glasgow J. Math. 34 (1992), 75–81. MR 1145633 |
| Reference:
|
[12] Jain S. K., López-Permouth S. R.: A survey on the theory of weakly injective modules.In: Computational Algebra, 205–233, Lecture notes in pure and applied mathematics, Marcel Dekker, Inc., New York, 1994. MR 1245954 |
| Reference:
|
[13] Kurshan A. P.: Rings whose cyclic modules have finitely generated socle.J. Algebra 14 (1970), 376–386. Zbl 0199.35503, MR 0260780 |
| Reference:
|
[14] López-Permouth S. R.: Rings characterized by their weakly injective modules.Glasgow Math. J. 34 (1992), 349–353. MR 1181777 |
| Reference:
|
[15] Malik S., Vanaja N.: Weak relative injective M-subgenerated modules.Advances in Ring Theory, Birkhauser, 1997, 221–239. Zbl 0934.16002, MR 1602677 |
| Reference:
|
[16] Page S., Zhou Y.: When direct sums of singular injectives are injective.In: Ring theory, Proceedings of the Ohio State-Denison Conference, World Scientific Publishing Co., 1993. Zbl 0853.16005, MR 1344237 |
| Reference:
|
[17] Page S., Zhou Y.: On direct sums of injective modules and chain conditions.Canad. J. Math. 46 (1994), 634–647. Zbl 0807.16006, MR 1276116 |
| Reference:
|
[18] Page S., Zhou Y.: Direct sums of quasi-injective modules, injective hulls, and natural classes.Comm. Alg. 22 (1994), 2911–2923. MR 1272360 |
| Reference:
|
[19] Saleh M.: A note on tightness.Glasgow Math. J. 41 (1999), 43–44. Zbl 0923.16003, MR 1689655 |
| Reference:
|
[20] Saleh M., Abdel-Mohsen A.: On weak injectivity and weak projectivity.In: Proceedings of the Mathematics Conference, World Scientific Press, New Jersey, 2000, 196–207. Zbl 0985.16002, MR 1773029 |
| Reference:
|
[21] Saleh M., Abdel-Mohsen A.: A note on weak injectivity.FJMS 11(2) (2003), 199–206. Zbl 1063.16004, MR 2020503 |
| Reference:
|
[22] Saleh M.: On q.f.d. modules and q.f.d. rings.Houston J. Math. 30 (2004), 629–636. Zbl 1070.16002, MR 2083867 |
| Reference:
|
[23] Saleh M.: On weakly projective and weakly injective modules.Comment. Math. Univ. Corolin. 45 (2004), 389–402. Zbl 1101.16004, MR 2103135 |
| Reference:
|
[24] Sanh N. V., Shum K. P., Dhompongsa S., Wongwai S.: On quasi-principally injective modules.Algebra Colloq. 6 (1999), 296–276. Zbl 0949.16003, MR 1809646 |
| Reference:
|
[25] Sanh N. V., Dhompongsa S., Wongwai S.: On generalized q.f.d. modules and rings.Algebra and Combinatorics (1999), 367–372. MR 1733193 |
| Reference:
|
[26] Wisbauer R.: Foundations of module and ring theory.Gordon and Breach, 1991. Zbl 0746.16001, MR 1144522 |
| Reference:
|
[27] Zhou Y.: Notes on weakly semisimple rings.Bull. Austral. Math. Soc. 52 (1996), 517–525. MR 1358705 |
| Reference:
|
[28] Zhou Y.: Weak injectivity and module classes.Comm. Algebra 25 (1997), 2395–2407. Zbl 0934.16004, MR 1459568 |
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