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A right continuous right weakly si-ring is semisimple

Published online by Cambridge University Press:  17 April 2009

Dinh Van Huynh
Institute of Mathematics, PO Box 631, Boho, Hanoi, Vietnam
Nguyen Van Sanh
Department of Mathematics, Hue University of Pedagogy, 32 Le Loi St, Hue, Vietnam
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It is shown that a projective CS right module M over a ring R is a direct sum of uniform modules of composition lengths at most 2 if (i) every finitely generated direct summand of M is continuous and (ii) every non-zero M-singular right R-module contains a non-zero M-injective submodule. In particular, a right continuous ring R is semisimple if R is right weakly SI, that is, if every non-zero singular right R-module contains a non-zero injective submodule.

Research Article
Copyright © Australian Mathematical Society 1995


[1]Anderson, F.W. and Fuller, K.R., Rings and categories of modules (Springer-Verlag, Berlin, Heidelberg, New York, 1974).Google Scholar
[2]Baccella, G., ‘Semiartinian V-rings and semiartinian von Neumann regular rings’, J. Algebra (to appear).Google Scholar
[3]Camillo, V. and Fuller, K.R., ‘On Loewy length of rings’, Pacific J. Math. 53 (1974), 347354.Google Scholar
[4]Chatters, A.W. and Hajarnavis, , Rings with chain conditions (Pitman, London, 1980).Google Scholar
[5]Cozzens, J. and Faith, C., Simple Noetherian rings (Cambridge University Press, London, 1985).Google Scholar
[6]Dung, N.V. and Smith, P.F., ‘On semi-artinian V-modules’, J. Pure Appl. Algebra 82 (1992), 2737.Google Scholar
[7]Dung, N.V., Huynh, D.V., Smith, P.F. and Wisbauer, R., Extending modules (Pitman, London, 1994).Google Scholar
[8]Faith, C., Algebra II: Ring theory (Springer-Verlag, Berlin, Heidelberg, New York, 1976).Google Scholar
[9]Goodearl, K.R., Singular torsion and the splitting properties (Memoirs of the Amer. Math. Soc, No 124, 1972).Google Scholar
[10]Huynh, D.V. and Wisbauer, R., ‘A structure theorem for SI-modules’, Glasgow Math. J. 34 (1992), 8389.Google Scholar
[11]Mohamed, S.H. and M¨ller, B.J., Continuous and discrete modules, London Math. Soc. Lecture Note Series 147 (Cambridge University Press, London, 1990).Google Scholar
[12]Osofsky, B.L., ‘Non-quasi-continuous quotients of finitely generated quasi-continuous modules’, in Ring Theory, Proceedings Denison Conference 1992 (World Scientific, Singapore, 1993), pp. 259275.Google Scholar
[13]Osofsky, B.L. and Smith, P.F., ‘Cyclic modules whose quotients have 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, Proceedings Conference Athens, OH (USA) 1989, Lecture Notes Math. 1448 (Springer-Verlag, Berlin, Heidelberg, New York, 1990), pp. 116124.Google Scholar
[15]Sanh, N.V., ‘On weakly SI-modules’, Bull. Austral. Math. Soc. 49 (1994), 159164.Google Scholar
[16]Smith, P.F., ‘Decomposing modules into projectives and injectives’, Pacific J. Math 78 (1978), 247266.Google Scholar
[17]Smith, P.F., ‘Rings characterized by their cyclic modules’, Canad. J. Math. 31 (1979), 93111.Google Scholar
[18]Stenström, B., Rings of quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[19]Wisbauer, R., Foundations of module and ring theory (Gordon and Breach, Reading, 1991).Google Scholar
[20]Yousif, M.F., ‘SI-modules’, Math. J. Okayama Univ. 28 (1986), 133146.Google Scholar