Geodesic
On generalized q.f.d. modules
Archivum mathematicum, Tome 41 (2005) no. 3, pp. 243-251 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 16D10, 16D50, 16D60, 16D70, 16D90
Keywords: tight; weakly tight; weakly injective; q.f.d.; generalized q.f.d. modules; generalized weakly semisimple 
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}
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Saleh, Mohammad; Jain, S. K.; López-Permouth, S. R. On generalized q.f.d. modules. Archivum mathematicum, Tome 41 (2005) no. 3, pp. 243-251. http://geodesic.mathdoc.fr/item/ARM_2005_41_3_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] Camillo V. P.: Modules whose quotients have finite Goldie dimension. Pacific J. Math. 69 (1977), 337–338. | MR | Zbl

[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

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

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

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

[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

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

[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

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

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

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

[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. | MR | Zbl

[17] Page S., Zhou Y.: On direct sums of injective modules and chain conditions. Canad. J. Math. 46 (1994), 634–647. | MR | Zbl

[18] Page S., Zhou Y.: Direct sums of quasi-injective modules, injective hulls, and natural classes. Comm. Alg. 22 (1994), 2911–2923. | MR

[19] Saleh M.: A note on tightness. Glasgow Math. J. 41 (1999), 43–44. | MR | Zbl

[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. | MR | Zbl

[21] Saleh M., Abdel-Mohsen A.: A note on weak injectivity. FJMS 11(2) (2003), 199–206. | MR | Zbl

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

[23] Saleh M.: On weakly projective and weakly injective modules. Comment. Math. Univ. Corolin. 45 (2004), 389–402. | MR | Zbl

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

[25] Sanh N. V., Dhompongsa S., Wongwai S.: On generalized q.f.d. modules and rings. Algebra and Combinatorics (1999), 367–372. | MR

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

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

[28] Zhou Y.: Weak injectivity and module classes. Comm. Algebra 25 (1997), 2395–2407. | MR | Zbl