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A note on GV-modules with Krull dimension

Published online by Cambridge University Press:  18 May 2009

Dinh Huynh van ,
Patrick F. Smith  and
Robert Wisbauer
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
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Abstract

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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 K ⊂ M 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
Information
Glasgow Mathematical Journal , Volume 32 , Issue 3 , September 1990 , pp. 389 - 390
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

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