Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-27T19:31:42.017Z Has data issue: false hasContentIssue false
  • English
  • Français

Decompositions of modules into projective modules and CS-modules

Published online by Cambridge University Press:  17 April 2009

Somyot Plubtieng
Somyot Plubtieng
Affiliation:
Department of Mathematics, Naresuan University, Phitsanulok 65000, Thailand e-mail: somyotp@nu.ac.th
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.

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 159 - 164
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Anderson, F.W. and Fuller, K.R., Rings and categories of modules (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[2]Clark, J. and Wisbauer, R., ‘Σ-extending modules’, J. Pure Appl. Algebra 104 (1995), 19–32.CrossRefGoogle Scholar
[3]Dung, N.V., ‘On indecomposable decompositions of CS-modules II’, J. Pure Appl. Algebra 119 (1997), 139–153.CrossRefGoogle Scholar
[4]Dung, N.V. and Smith, P.F., ‘Rings for which certain modules are CS’, J. Pure Appl. Algebra 102 (1995), 273–287.CrossRefGoogle Scholar
[5]Dung, N.V., Huynh, D.V., Smith, P.F. and Wisbauer, R., Extending modules, Pitman Research Notes in Mathematics Series 313 (Longman, Harlow, 1994).Google Scholar
[6]Faith, C., Algebra II: Ring theory (Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[7]Garcia, J.L. and Dung, N.V., ‘Some decomposition properties of injective and pure-injective modules’, Osaka J. Math. 31 (1994), 95–108.Google Scholar
[8]Garcia, J.L. and Hernandez, J. Martinez, ‘Purity through Gabriel's functor rings’, Bull. Soc. Math. Belgique 31 (1994), 95–108.Google Scholar
[9]Huynh, D.V., Rizvi, S.T. and Yousif, M.F., ‘Rings whose finitely generated modules are extending’, J. Pure Appl. Algebra 111 (1996), 325–328.CrossRefGoogle Scholar
[10]Huynh, D.V. and Rizvi, S.T., ‘On some classes of artinian rings’, J. Algebra (to appear).Google Scholar
[11]Mohamed, S.H. and Müller, B.J., Continuous and discrete modules, London Math. Soc. Lecture Notes 147 (Cambridge Univ. Press., Cambridge, 1990).CrossRefGoogle Scholar
[12]Vanaja, N., ‘All finitely generated M-subgenerated modules are extending’, Comm. Algebra 24 (1996), 543–572.CrossRefGoogle Scholar
[13]Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Philadelphia, P.A., 1991).Google Scholar