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Some remarks on CS modules and SI rings

Published online by Cambridge University Press:  17 April 2009

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

Research Article
Copyright © Australian Mathematical Society 2002


[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), 2529.Google Scholar
[3]Clark, J. and Wisbauer, R., ‘Σ-extending modules’, J. Pure Appl. Algebra 104 (1995), 1932.Google Scholar
[4]Dung, N.V. and Smith, P.F., ‘Rings for which certain modules are CS’, J. Pure Appl. Algebra 102 (1995), 257265.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).Google 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), 31313139.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), 42194222.Google Scholar
[10]Oshiro, K., ‘Lifting modules, extending modules and their applications to QF rings’, Hokkaido Math. J. 13 (1984), 310338.Google Scholar
[11]Oshiro, K., ‘On Harada rings, I, II, III’, Math. J. Okayama Univ. 31 (1989), 161178, 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. 279301.Google Scholar
[13]Osofsky, B.L. and Smith, P.F., ‘Cyclic modules whose quotients have all complement submodules direct summands’, J. Algebra 139 (1991), 342354.Google 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. 116124.Google Scholar