| Title:
|
On weakly projective and weakly injective modules (English) |
| Author:
|
Saleh, Mohammad |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
389-402 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module $M$, there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly injective in $\sigma [M]$, for any $N\in \sigma [M]$. Similarly, if $M$ is projective and right perfect in $\sigma [M]$, then there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly projective in $\sigma [M]$, for any $N\in \sigma [M]$. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes $\Cal M$ of modules in $\sigma [M]$, we study when direct sums of modules from $\Cal M$ satisfy property $\Bbb P$ in $\sigma [M]$. In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules. (English) |
| Keyword:
|
tight |
| Keyword:
|
weakly tight |
| Keyword:
|
weakly injective |
| Keyword:
|
weakly projective |
| Keyword:
|
countably thick |
| Keyword:
|
locally q.f.d. |
| Keyword:
|
weakly semisimple |
| MSC:
|
16D40 |
| MSC:
|
16D50 |
| MSC:
|
16D60 |
| MSC:
|
16D70 |
| MSC:
|
16D90 |
| MSC:
|
16P40 |
| idZBL:
|
Zbl 1101.16004 |
| idMR:
|
MR2103135 |
| . |
| Date available:
|
2009-05-05T16:46:03Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119468 |
| . |
| Reference:
|
[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] Brodskii G.: Denumerable distributivity, linear compactness and the AB5$^{\ast }$ condition in modules.Russian Acad. Sci. Dokl. Math. 53 (1996), 76-77. |
| Reference:
|
[6] Brodskii G.: The Grothendieck condition AB5$^{\ast }$ and generalizations of module distributivity.Russ. Math. 41 (1997), 1-11. MR 1480764 |
| Reference:
|
[7] Camillo V.P.: Modules whose quotients have finite Goldie dimension.Pacific J. Math. 69 (1977), 337-338. Zbl 0356.13003, MR 0442020 |
| Reference:
|
[8] Dung N.V., Huynh D.V., Smith P.F., Wisbauer R.: Extending Modules.Pitman, London, 1994. Zbl 0841.16001 |
| Reference:
|
[9] 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:
|
[10] Goel V.K., Jain S.K.: $\pi $-injective modules and rings whose cyclic modules are $\pi $-injective.Comm. Algebra 6 (1978), 59-73. MR 0491819 |
| Reference:
|
[11] Golan J.S., López-Permouth S.R.: QI-filters and tight modules.Comm. Algebra 19 (1991), 2217-2229. MR 1123120 |
| Reference:
|
[12] 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:
|
[13] Jain S.K., López-Permouth S.R., Risvi T.: A characterization of uniserial rings via continuous and discrete modules.J. Austral. Math. Soc., Ser. A 50 (1991), 197-203. MR 1094917 |
| Reference:
|
[14] Jain S.K., López-Permouth S.R., Saleh M.: On weakly projective modules.in: Ring Theory, Proceedings, OSU-Denison conference 1992, World Scientific Press, New Jersey, 1993, pp.200-208. MR 1344231 |
| Reference:
|
[15] 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:
|
[16] 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:
|
[17] Jain S.K., López-Permouth S.R.: A survey on the theory of weakly injective modules.in: Computational Algebra, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1994, pp.205-233. MR 1245954 |
| Reference:
|
[18] Kurshan A.P.: Rings whose cyclic modules have finitely generated socle.J. Algebra 14 (1970), 376-386. Zbl 0199.35503, MR 0260780 |
| Reference:
|
[19] López-Permouth S.R.: Rings characterized by their weakly injective modules.Glasgow Math. J. 34 (1992), 349-353. MR 1181777 |
| Reference:
|
[20] Malik S., Vanaja N.: Weak relative injective $M$-subgenerated modules.Advances in Ring Theory, Birkhäuser, Boston, 1997, pp.221-239. Zbl 0934.16002, MR 1602677 |
| Reference:
|
[21] Mohamed S., Muller B., Singh S.: Quasi-dual continuous modules.J. Austral. Math. Soc., Ser. A 39 (1985), 287-299. MR 0802719 |
| Reference:
|
[22] Mohamed S., Muller B, HASH(0xa277a48): Continuous and Discrete Modules.Cambridge University Press, 1990. MR 1084376 |
| Reference:
|
[23] Saleh M.: A note on tightness.Glasgow Math. J. 41 (1999), 43-44. Zbl 0923.16003, MR 1689655 |
| Reference:
|
[24] Saleh M., Abdel-Mohsen A.: On weak injectivity and weak projectivity.in: Proceedings of the Mathematics Conference, World Scientific Press, New Jersey, 2000, pp.196-207. Zbl 0985.16002, MR 1773029 |
| Reference:
|
[25] Saleh M., Abdel-Mohsen A.: A note on weak injectivity.Far East J. Math. Sci. 11 (2003), 199-206. Zbl 1063.16004, MR 2020503 |
| Reference:
|
[26] Saleh M.: On q.f.d. modules and q.f.d. rings.Houston J. Math., to appear. Zbl 1070.16002, MR 2083867 |
| Reference:
|
[27] 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:
|
[28] Sanh N.V., Dhompongsa S., Wongwai S.: On generalized q.f.d. modules and rings.Algebra and Combinatorics, Springer-Verlag, 1999, pp.367-272. MR 1733193 |
| Reference:
|
[29] Wisbauer R.: Foundations of Module and Ring Theory.Gordon and Breach, 1991. Zbl 0746.16001, MR 1144522 |
| Reference:
|
[30] Zhou Y.: Notes on weakly semisimple rings.Bull. Austral. Math. Soc. 52 (1996), 517-525. MR 1358705 |
| Reference:
|
[31] Zhou Y.: Weak injectivity and module classes.Comm. Algebra 25 (1997), 2395-2407. Zbl 0934.16004, MR 1459568 |
| . |
