Geodesic
Generalized injectivity and chain conditions
Glasgow mathematical journal, Tome 34 (1992) no. 3, pp. 319-326

Voir la notice de l'article provenant de la source Cambridge University Press

Relationships between injectivity or generalized injectivity and chain conditions on a module category have been studied by several authors. A well-known theorem of Osofsky [14, 15] asserts that a ring all of whose cyclic right modules are injective is semisimple Artinian. Osofsky's proofs in [14, 15] essentially used homological properties of injective modules, and, later, her arguments were applied by other authors in their studies of rings for which cyclic right modules are quasi-injective, continuous or quasi-continuous (see e. g. [1, 10, 12]). Following [5] (cf. [4]), a module M is called a CS-module if every submodule of M is essential in a direct summand of M. In the recent paper [17], B. L. Osofsky and P. F. Smith have proved a very general theorem on cyclic completely CS-modules from which many known results in this area follow rather easily. In another direction, it was proved in [8] that a finitely generated quasi-injective module with ACC (respectively DCC) on essential submodules is Noetherian (respectively Artinian). This result was also extended to CS-modules in [3, 16], and weak CS-modules in [19]. 
Dung, Nguyen V. Generalized injectivity and chain conditions. Glasgow mathematical journal, Tome 34 (1992) no. 3, pp. 319-326. doi: 10.1017/S0017089500008880
@article{10_1017_S0017089500008880,
     author = {Dung, Nguyen V.},
     title = {Generalized injectivity and chain conditions},
     journal = {Glasgow mathematical journal},
     pages = {319--326},
     year = {1992},
     volume = {34},
     number = {3},
     doi = {10.1017/S0017089500008880},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008880/}
}
TY  - JOUR
AU  - Dung, Nguyen V.
TI  - Generalized injectivity and chain conditions
JO  - Glasgow mathematical journal
PY  - 1992
SP  - 319
EP  - 326
VL  - 34
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008880/
DO  - 10.1017/S0017089500008880
ID  - 10_1017_S0017089500008880
ER  - 
%0 Journal Article
%A Dung, Nguyen V.
%T Generalized injectivity and chain conditions
%J Glasgow mathematical journal
%D 1992
%P 319-326
%V 34
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500008880/
%R 10.1017/S0017089500008880
%F 10_1017_S0017089500008880

[1] 1.Ahsan, J., Rings all of whose cyclic modules are quasi-injective. Proc. London Math Soc. (3) 27 (1973), 425–439. Google Scholar | DOI

[2] 2.Armendariz, E. P., Rings with dec on essential left ideals. Comm. Algebra 8 (1980), 299–308. Google Scholar | DOI

[3] 3.Camillo, V. and Yousif, M. F., CS-modules with ace or dec on essential submodules, Comm. Algebra 19 (1991), 655–662. Google Scholar

[4] 4.Chatters, A. W. and Hajarnavis, C. R., Rings in which every complement right ideal is a direct summand. Quart. J. Math. Oxford (2) 28 (1977), 61–80. Google Scholar | DOI

[5] 5.Chatters, A. W. and Khuri, S. M., Endomorphism rings of modules over non-singular CS-rings. J. London Math. Soc. (2) 21 (1980), 434–444. Google Scholar | DOI

[6] 6.Van Huynh, Dinh and Dung, Nguyen V., A characterization of Artinian rings. Glasgow Math. J. 30 (1988), 67–73. Google Scholar | DOI

[7] 7.Van Huynh, Dinh, Dung, Nguyen V. and Smith, P. F., A characterization of rings with Krull dimension. J. Algebra 132 (1990), 104–112. Google Scholar | DOI

[8] 8.Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., Quasi-injective modules with ace or dec on essential submodules. Arch. Math. 53 (1989), 252–255. Google Scholar

[9] 9.Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., On modules with finite uniform and Krull dimension. Arch. Math. 57 (1991), 122–132. Google Scholar | DOI

[10] 10.Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are π-injective. Comm. Algebra 6 (1978), 59–73. Google Scholar | DOI

[11] 11.Jain, S. K., López-Permouth, S. R. and Rizvi, S. T., Continuous rings with ace on essentials are Artinian. Proc. Amer. Math. Soc. 108 (1990), 583–586. Google Scholar | DOI

[12] 12.Jain, S. K. and Mohamed, S., Rings whose cyclic modules are continuous. Indian Math. Soc. Journal 42 (1978), 197–202. Google Scholar

[13] 13.Dung, Nguyen V., Modules whose closed submodules are finitely generated. Proc. Edinburgh Math. Soc. 34 (1991), 161–166. Google Scholar | DOI

[14] 14.Osofsky, B. L., Rings all of whose finitely generated modules are injective. Pacific J. Math. 14 (1964), 645–650. Google Scholar | DOI

[15] 15.Osofsky, B. L., Noninjective cyclic modules. Proc. Amer. Math. Soc. 19 (1968), 1383–1384. Google Scholar | DOI

[16] 16.Osofsky, B. L., Chain conditions on essential submodules. Proc. Amer. Math. Soc. 114 (1992), 11–19. Google Scholar | DOI

[17] 17.Osofsky, B. L. and Smith, P. F., Cyclic modules whose quotients have all complement submodules direct summands. J. Algebra, 139 (1991), 342–354. Google Scholar | DOI

[18] 18.Page, S. S. and Yousif, M. F., Relative injectivity and chain conditions. Comm. Algebra 17 (1989), 899–924. Google Scholar | DOI

[19] 19.Smith, P. F., CS-modules and weak CS-modules, in Non-commutative ring theory, Lecture Notes in Mathematics No. 1448 (Springer-Verlag, 1990), 99–115. Google Scholar | DOI

Cité par Sources :