Skip to main content Accessibility help
×
×
Home

A note on GV-modules with Krull dimension

  • Dinh Huynh van (a1), Patrick F. Smith (a2) and Robert Wisbauer (a3)
Abstract

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]).

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A note on GV-modules with Krull dimension
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A note on GV-modules with Krull dimension
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A note on GV-modules with Krull dimension
      Available formats
      ×
Copyright
References
Hide All
1.Boyle, A. K. and Goodearl, K. R., Rings over which certain modulesare injective, Pacific J. Math. 58 (1975), 4353.
2.Fuller, K., Relative projectivity and injectivity classes determined by simple modules, J. London Math. Soc. 5 (1972), 423431.
3.Hirano, Y., Regular modules and V-modules, Hiroshima Math. J. 11 (1981), 125142.
4.Page, S. S. and Yousif, M. F., Relative injectivity and chain conditions, Comm. Algebra, 17 (1989), 899924.
5.Ramamurthi, V. S. and Rangaswamy, K. M., Generalized V-rings, Math. Scand. 31 (1972), 6977.
6.Shock, R. C., Dual generalization of the artinian and noetherian conditions, Pacific J. Math. 54 (1974), 227235.
7.Tominaga, H., On s-unital rings, Math. J. Okayama Univ. 18 (1976), 117134.
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.
9.Wisbauer, R., Co-semisimple modules and nonassociative V-rings, Comm. Algebra 5 (1977), 11931209.
10.Wisbauer, R., Grundlagen der Modul- und Ringtheorie, (Verlag R. Fischer, München 1988).
11.Yousif, M. F., V-modules with Krull dimension, Bull. Austral. Math. Soc. 37 (1988), 237240.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed