Geodesic
Countably thick modules
Archivum mathematicum, Tome 41 (2005) no. 4, pp. 349-358 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The purpose of this paper is to further the study of countably thick modules via weak injectivity. Among others, for some classes ${\mathcal M}$ of modules in $ \sigma [M]$ we study when direct sums of modules from ${\mathcal M}$ satisfies a property $\mathbb P$ in $\sigma [M]$. In particular, we get characterization of locally countably thick modules, a generalization of locally q.f.d. modules.
The purpose of this paper is to further the study of countably thick modules via weak injectivity. Among others, for some classes ${\mathcal M}$ of modules in $ \sigma [M]$ we study when direct sums of modules from ${\mathcal M}$ satisfies a property $\mathbb P$ in $\sigma [M]$. In particular, we get characterization of locally countably thick modules, a generalization of locally q.f.d. modules.
Classification : 16D50, 16D60, 16D70, 16D90
Keywords: tight; weakly tight; weakly injective; countably thick; locally q.f.d.; weakly semisimple 
@article{ARM_2005_41_4_a0,
     author = {Abdel-Mohsen, Ali and Saleh, Mohammad},
     title = {Countably thick modules},
     journal = {Archivum mathematicum},
     pages = {349--358},
     year = {2005},
     volume = {41},
     number = {4},
     mrnumber = {2195489},
     zbl = {1114.16003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a0/}
}
TY  - JOUR
AU  - Abdel-Mohsen, Ali
AU  - Saleh, Mohammad
TI  - Countably thick modules
JO  - Archivum mathematicum
PY  - 2005
SP  - 349
EP  - 358
VL  - 41
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a0/
LA  - en
ID  - ARM_2005_41_4_a0
ER  - 
%0 Journal Article
%A Abdel-Mohsen, Ali
%A Saleh, Mohammad
%T Countably thick modules
%J Archivum mathematicum
%D 2005
%P 349-358
%V 41
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a0/
%G en
%F ARM_2005_41_4_a0
Abdel-Mohsen, Ali; Saleh, Mohammad. Countably thick modules. Archivum mathematicum, Tome 41 (2005) no. 4, pp. 349-358. http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a0/

[1] Albu T., Nastasescu C.: Relative finiteness in module theory. Marcel Dekker 1984. | MR | Zbl

[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

[3] Berry D.: Modules whose cyclic submodules have finite dimension. Canad. Math. Bull. 19 (1976), 1–6. | MR | Zbl

[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

[5] Brodskii G.: Denumerable distributivity, linear compactness and the AB5$^{\ast }$ condition in modules. Russian Acad. Sci. Dokl. Math. 53 (1996), 76–77.

[6] Brodskii G.: The Grothendieck condition AB5$^{\ast }$ and generalizations of module distributivity. Russ. Math. 41 (1997), 1–11. | MR

[7] Camillo V. P.: Modules whose quotients have finite Goldie dimension. Pacific J. Math. 69 (1977), 337–338. | MR | Zbl

[8] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R.: Extending modules. Pitman, London, 1994. | Zbl

[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

[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

[11] Golan J. S., López-Permouth S. R.: QI-filters and tight modules. Comm. Algebra 19 (1991), 2217–2229. | MR

[12] Jain S. K., López-Permouth S. R.: Rings whose cyclics are essentially embeddable in projective modules. J. Algebra 128 (1990), 257–269. | MR

[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

[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, 200–208. | MR

[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

[16] Jain S. K., López-Permouth S. R., Singh S.: On a class of QI-rings. Glasgow J. Math. 34 (1992), 75–81. | MR

[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, 205–233. | MR

[18] Kurshan A. P.: Rings whose cyclic modules have finitely generated socle. J. Algebra 14 (1970), 376–386. | MR | Zbl

[19] López-Permouth S. R.: Rings characterized by their weakly injective modules. Glasgow Math. J. 34 (1992), 349–353. | MR

[20] Malik S., Vanaja N.: Weak relative injective M-subgenerated modules. Advances in Ring Theory, Birkhauser, 1997, 221–239. | MR | Zbl

[21] Mohamed S., Muller B., Singh S.: Quasi-dual continuous modules. J. Austral. Math. Soc., Ser. A 39 (1985), 287–299. | MR

[22] Mohamed S., Muller B.: Continuous and discrete modules. Cambridge University Press 1990. | MR

[23] Saleh M.: A note on tightness. Glasgow Math. J. $\mathbf{41}$ (1999), 43–44. | MR | Zbl

[24] 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. | MR | Zbl

[25] Saleh M., Abdel-Mohsen A.: A note on weak injectivity. Far East Journal of Mathematical Sciences (FJMS) 11 (2003), 199-20-6. | MR | Zbl

[26] Saleh M.: On q.f.d. modules and q.f.d. rings. Houston J. Math. 30 (2004), 629–636. | MR | Zbl

[27] Sanh N. V., Shum K. P., Dhompongsa S., Wongwai S.: On quasi-principally injective modules. Algebra Colloq. 6 (1999), 296–276. | MR | Zbl

[28] Sanh N. V., Dhompongsa S., Wongwai S.: On generalized q.f.d. modules and rings. Algebra and Combinatorics, Springer-Verlag, 1999, 367–272. | MR

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

[30] Zhou Y.: Notes on weakly semisimple rings. Bull. Austral. Math. Soc. 52 (1996), 517–525. | MR

[31] Zhou Y.: Weak injectivity and module classes. Comm. Algebra $\mathbf{25}$ (1997), 2395–2407. | MR | Zbl