Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-22T07:46:43.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Some remarks on CS modules and SI rings

Published online by Cambridge University Press:  17 April 2009

Dinh Van Huynh
Dinh Van Huynh
Affiliation:
Department of Mathematics, Ohio University, Athens OH 45701, United States of America, e-mail: [email protected]
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.

We discuss some results on CS modules, SI rings, and SC rings. Then we consider the question of when, over a right SC ring R, every right R-module is CS. In Theorem 3.1 we show that this is the case if and only if R is a right countably Σ-CS ring. In light of this, we give an example showing that a result proved by Chen (2000) is incorrect. Furthermore, Theorem 4.1 shows that the assumptions of Chen (2000) can be weakened considerably.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 3 , June 2002 , pp. 461 - 466
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Anderson, F.W. and Fuller, F.R., Rings and categories of modules (2nd Edition), Graduate Texts in Mathematics 13 (Springer-Verlag, New York, 1992).Google Scholar
[2]Chen, M.S., ‘On CS rings and QF rings’, Southeast Asian Bull. Math. 24 (2000), 25–29.CrossRefGoogle Scholar
[3]Clark, J. and Wisbauer, R., ‘Σ-extending modules’, J. Pure Appl. Algebra 104 (1995), 19–32.CrossRefGoogle Scholar
[4]Dung, N.V. and Smith, P.F., ‘Rings for which certain modules are CS’, J. Pure Appl. Algebra 102 (1995), 257–265.Google Scholar
[5]Dung, N.V., Huynh, D.V., Smith, P.F. and Wisbauer, R., Extending modules, Pitman Research Notes in Mathematics Series 313 (Longman Scientific and Technical, Harlow, 1994).Google Scholar
[6]Faith, C., Algebra II: Ring theory (Springer Verlag, Berlin, New York, 1976).CrossRefGoogle Scholar
[7]Goodearl, K.R., Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124 (American Mathematical Society, Providence, R.I., 1972).Google Scholar
[8]Huynh, D.V., ‘A right countably Σ-CS ring with acc or dcc on projective principal right ideals is left artinian and QF-3’, Trans. Amer. Math. Soc. 347 (1995), 3131–3139.Google Scholar
[9]Huynh, D.V., Jain, S.K. and López-Permough, S.R., ‘Rings characterized by direct sums of CS modules’, Comm. Algebra 28 (2000), 4219–4222.CrossRefGoogle Scholar
[10]Oshiro, K., ‘Lifting modules, extending modules and their applications to QF rings’, Hokkaido Math. J. 13 (1984), 310–338.Google Scholar
[11]Oshiro, K., ‘On Harada rings, I, II, III’, Math. J. Okayama Univ. 31 (1989), 161–178, 179–188; 32 (1990), 111–118.Google Scholar
[12]Oshiro, K., ‘Theories of Harada in artinian rings and applications to classical artinian rings’, in International Symposium on Ring Theory, (Birkenmeier, G.F., Park, J.K., Park, S., Editors), Trends Math. (Birkhäuser, Boston, 2001), pp. 279–301.CrossRefGoogle Scholar
[13]Osofsky, B.L. and Smith, P.F., ‘Cyclic modules whose quotients have all complement submodules direct summands’, J. Algebra 139 (1991), 342–354.CrossRefGoogle Scholar
[14]Rizvi, S.T. and Yousif, M.F., ‘On continuous and singular modules’, in Non-commutative Ring Theory, (Jain, S.K. and López-Permouth, S.R., Editors), Lecture Notes in Math. 1448 (Springer Verlag, Berlin, 1990), pp. 116–124.Google Scholar