Hostname: page-component-5d59c44645-zlj4b Total loading time: 0 Render date: 2024-02-22T05:15:35.061Z Has data issue: false hasContentIssue false

A note on GV-modules with Krull dimension

Published online by Cambridge University Press:  18 May 2009

Dinh Huynh van
Affiliation:
Institute of MathematicsP.O. Box 631 Bo hoHanoi, Vietnam
Patrick F. Smith
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QwScotland, U.K.
Robert Wisbauer
Affiliation:
Mathematisches Institutder Universität Düsseldorf4000 DüsseldorfFederal Republic ofGermany
Rights & Permissions [Opens in a new window]

Abstract

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.

Extending a result of Boyle and Goodearl in [1] on V-rings it was shown in Yousif [11] that a generalized V-module (GV-module) has Krull dimension if and only if it is noetherian. Our note is based on the observation that every GV-module has a maximal submodule (Lemma 1). Applying a theorem of Shock [6] we immediately obtain that a GV-module has acc on essential submodules if and only if for every essential submodule KM the factor module M/K has finitely generated socle. Yousif's result is obtained as a corollary.

Let R be an associative ring with unity and R-Mod the category of unital left R-modules. Soc M denotes the socle of an R-module M. If K ⊂ M is an essential submodule we write K⊴M.

An R-module M is called co-semisimple or a V-module, if every simple module is M-injective ([2], [7], [9], [10]). According to Hirano [3] M is a generalized V-module or GV-module, if every singular simple R-module is M-injective. This extends the notion of a left GV-ring in Ramamurthi-Rangaswamy [5].

It is easy to see that submodules, factor modules and direct sums of co-semisimple modules (GV-modules) are again co-semisimple (GV-modules) (e.g. [10, § 23]).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Boyle, A. K. and Goodearl, K. R., Rings over which certain modulesare injective, Pacific J. Math. 58 (1975), 4353.Google Scholar
2.Fuller, K., Relative projectivity and injectivity classes determined by simple modules, J. London Math. Soc. 5 (1972), 423431.Google Scholar
3.Hirano, Y., Regular modules and V-modules, Hiroshima Math. J. 11 (1981), 125142.Google Scholar
4.Page, S. S. and Yousif, M. F., Relative injectivity and chain conditions, Comm. Algebra, 17 (1989), 899924.Google Scholar
5.Ramamurthi, V. S. and Rangaswamy, K. M., Generalized V-rings, Math. Scand. 31 (1972), 6977.Google Scholar
6.Shock, R. C., Dual generalization of the artinian and noetherian conditions, Pacific J. Math. 54 (1974), 227235.Google Scholar
7.Tominaga, H., On s-unital rings, Math. J. Okayama Univ. 18 (1976), 117134.Google Scholar
8.van Huynh, D., Dung, N. V. and Wisbauer, R., Quasi-injective modules with ace or dcc on essential submodules, Arch. Math. (Basel) 53 (1989), 252255.Google Scholar
9.Wisbauer, R., Co-semisimple modules and nonassociative V-rings, Comm. Algebra 5 (1977), 11931209.Google Scholar
10.Wisbauer, R., Grundlagen der Modul- und Ringtheorie, (Verlag R. Fischer, München 1988).Google Scholar
11.Yousif, M. F., V-modules with Krull dimension, Bull. Austral. Math. Soc. 37 (1988), 237240.Google Scholar