Hostname: page-component-5d59c44645-dknvm Total loading time: 0 Render date: 2024-02-29T12:02:52.228Z Has data issue: false hasContentIssue false

Generalized injectivity and chain conditions

Published online by Cambridge University Press:  18 May 2009

Nguyen V. Dung
Departamento de Matemáticas, Universidad de Murcia, 30071 Murcia, Spain
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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].

Research Article
Copyright © Glasgow Mathematical Journal Trust 1992



1.Ahsan, J., Rings all of whose cyclic modules are quasi-injective. Proc. London Math Soc. (3) 27 (1973), 425439.Google Scholar
2.Armendariz, E. P., Rings with dec on essential left ideals. Comm. Algebra 8 (1980), 299308.Google Scholar
3.Camillo, V. and Yousif, M. F., CS-modules with ace or dec on essential submodules, Comm. Algebra 19 (1991), 655662.Google Scholar
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), 6180.Google Scholar
5.Chatters, A. W. and Khuri, S. M., Endomorphism rings of modules over non-singular CS-rings. J. London Math. Soc. (2) 21 (1980), 434444.Google Scholar
6.Van Huynh, Dinh and Dung, Nguyen V., A characterization of Artinian rings. Glasgow Math. J. 30 (1988), 6773.Google Scholar
7.Van Huynh, Dinh, Dung, Nguyen V. and Smith, P. F., A characterization of rings with Krull dimension. J. Algebra 132 (1990), 104112.Google Scholar
8.Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., Quasi-injective modules with ace or dec on essential submodules. Arch. Math. 53 (1989), 252255.Google Scholar
9.Van Huynh, Dinh, Dung, Nguyen V. and Wisbauer, R., On modules with finite uniform and Krull dimension. Arch. Math. 57 (1991), 122132.Google Scholar
10.Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are π-injective. Comm. Algebra 6 (1978), 5973.Google Scholar
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), 583586.Google Scholar
12.Jain, S. K. and Mohamed, S., Rings whose cyclic modules are continuous. Indian Math. Soc. Journal 42 (1978), 197202.Google Scholar
13.Dung, Nguyen V., Modules whose closed submodules are finitely generated. Proc. Edinburgh Math. Soc. 34 (1991), 161166.Google Scholar
14.Osofsky, B. L., Rings all of whose finitely generated modules are injective. Pacific J. Math. 14 (1964), 645650.Google Scholar
15.Osofsky, B. L., Noninjective cyclic modules. Proc. Amer. Math. Soc. 19 (1968), 13831384.Google Scholar
16.Osofsky, B. L., Chain conditions on essential submodules. Proc. Amer. Math. Soc. 114 (1992), 1119.Google Scholar
17.Osofsky, B. L. and Smith, P. F., Cyclic modules whose quotients have all complement submodules direct summands. J. Algebra, 139 (1991), 342354.Google Scholar
18.Page, S. S. and Yousif, M. F., Relative injectivity and chain conditions. Comm. Algebra 17 (1989), 899924.Google Scholar
19.Smith, P. F., CS-modules and weak CS-modules, in Non-commutative ring theory, Lecture Notes in Mathematics No. 1448 (Springer-Verlag, 1990), 99115.Google Scholar