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Simple rings with injectivity conditions on one-sided ideals

Published online by Cambridge University Press:  17 April 2009

John Clark
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand e-mail:
Dinh Van Huynh
Department of Mathematics, Ohio University, Athens, OH. 45701, United States of America e-mail:
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This paper looks at simple rings which have right ideals satisfying various types of injectivity conditions. We characterise when a simple regular ring is right self-injective and show that if R is a simple ring in which every right ideal is the direct sum of quasi-continuous right ideals then R is either Artinian or a non-selfinjective right Goldie ring in which every right ideal is a direct sum of uniform right ideals.

Research Article
Copyright © Australian Mathematical Society 2007


[1]Anderson, F.W. and Fuller, K.R., Rings and categories of modules, (second edition) (Springer-Verlag, New York, 1992).Google Scholar
[2]Chatters, A.W. and Hajarnavis, C.R., ‘Rings in which every complement right ideal is a direct summand’, Quart. J. Math. Oxford 28 (1977), 6180.Google Scholar
[3]Clark, J. and Huynh, D.V., ‘A study of uniform one-sided ideals in simple rings’, Glasgow Math. J. (to appear).Google Scholar
[4]Cozzens, J.H., ‘Homological properties of the ring of differential polynomials’, Bull. Amer. Math. Soc. 76 (1970), 7579.Google Scholar
[5]Dinh, H.Q., ‘A note on pseudo-injective modules’, Comm. Algebra 33 (2005), 361369.Google Scholar
[6]Dung, N.V., Huynh, D.V., Smith, P.F. and Wisbauer, R., Extending modules (Longman Scientific & Technical, Harlow, 1994).Google Scholar
[7]Dung, N.V. and Smith, P.F., ‘Σ-CS-modules’, Comm. Algebra 22 (1994), 8393.Google Scholar
[8]Goodearl, K.R., Von Neumann regular rings, (second edition) (Krieger Publishing Company, Malabar, 1991).Google Scholar
[9]Goodearl, K.R. and Handelman, D., ‘Simple self-injective rings’, Comm. Algebra 3 (1975), 797834.Google Scholar
[10]Hart, R., ‘Simple rings with uniform right ideals’, J. London Math. Soc. Ser. 1 42 (1967), 614617.Google Scholar
[11]Huynh, D.V., Jain, S.K. and López-Permouth, S.R., ‘Prime Goldie rings of uniform dimension at least two and with all one-sided ideals CS are semihereditary’, Comm. Algebra 31 (2003), 53555360.Google Scholar
[12]Lam, T.Y., Lectures on modules and rings (Springer-Verlag, New York, 1999).Google Scholar
[13]Mohamed, S.H. and Müller, B.J., Continuous and discrete modules (Cambridge Univ. Press, Cambridge, 1990).Google Scholar
[14]Osofsky, B.L., ‘Rings all of whose finitely generated modules are injective’, Pacific J. Math. 14 (1964), 645650.Google Scholar
[15]Utumi, Y., ‘On continuous rings and self injective rings’, Trans. Amer. Math. Soc. 118 (1965), 158173.Google Scholar